OCEAN ENGINEERING GROUP

REPORT No.

12-02

OCEAN ENGINEERING GROUP

Three Dimensional Viscous/Inviscid Interactive Method

and Its Application to Propeller Blades

Xiangming Yu

August 2012

ENVIRONMENTAL AND WATER RESOURCES ENGINEERING

DEPARTMENT OF CIVIL, ARCHITECTURAL AND ENVIRONMENTAL ENG.

THE UNIVERSITY OF TEXAS AT AUSTIN

AUSTIN, TX 78712

Copyright

by

Xiangming Yu

2012

The Thesis Committee for Xiangming Yucertifies that this is the approved version of the following thesis:

Three Dimensional Viscous/Inviscid Interactive

Method and Its Application to Propeller Blades

Committee:

Spyros A. Kinnas, Supervisor

Howard Liljestrand



by

Xiangming Yu, B.E.

THESIS

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE in ENGINEERING

THE UNIVERSITY OF TEXAS AT AUSTIN

August 2012

Dedicated to my family and friends.

Acknowledgments

I first wish to thank my supervisor, Prof. Kinnas, for his continued

guidance and assistance over the last two years. His extraordinary wisdom

and unlimited passion for research inspires me to pursue my own academic

career. His encouragements and suggestions lead me forward, and help me

overcome difficulties in both my research and life. I do and will benefit from

him all my life. Also I want to thank Prof. Liljestrand for his reviewing of my

thesis.

Then I would like to thank all my friends in the Computational Hy-

drodynamics Laboratory (CHL) for their selfless help and companionship. I

would especially thank Mr. Ye Tian, Shu-Hao Chang, Liwei Han and Dr. Lei

He for their significant help not only in my research, but also in many aspects

of my life.

I would also thank my friends at Austin: Mr. Wentao Fu, Xudong Xu,

Haizhao Yang, Shichao Dai and Miss Xiaoqian Li. My life at Austin becomes

more colorful and memorable because of their companionship.

Finally, I want to express my sincere appreciation to my parents, Xi-

aosheng Yu and Guixia Liu. Without their love and encouragement, I would

never finish this work.

This work was supported by the U.S. Office of Naval Research (Con-

v

tracts N00014-07-1-0616 and N00014-10-1-0931) and members of the Phases V

and VI of the Consortium on Cavitation Performance of High Speed Propul-

sors: American Bureau of Shipping, Kawasaki Heavy Industry Ltd., Rolls-

Royce Marine AB, Rolls-Royce Marine AS, SSPA AB, Andritz Hydro GmbH,

Wartsilaa Netherlands B.V., Wartsilaa Norway AS, Wartsilaa Lips Defense

S.A.S., Wartsilaa CME Zhenjiang Propeller Co. Ltd, Daewoo Shipbuilding &

Marine Engineering Co., Ltd. and Samsung Heavy Industries Co., Ltd.

vi



Publication No.

Xiangming Yu, M.S.E.

The University of Texas at Austin, 2012

Supervisor: Spyros A. Kinnas

A three dimensional viscous/inviscid interactive boundary layer method

for predicting the effects of fluid viscosity on the performance of fully wetted

propellers is presented. This method is developed by coupling a three dimen-

sional low-order potential based panel method and a two dimensional integral

boundary layer analysis method. To simplify the solution procedures, this

method applies a reasonable assumption that the effects of the boundary lay-

er along the span wise direction (radially outward for propeller blades) could

be negligible compared with those along the stream wise direction (constant

radius for propeller blades). One significant development of this method, com-

pared with previous work, is to completely consider the effects of the added

sources on the whole blades and wakes rather than evaluate the boundary lay-

er effects along each strip, without interaction among strips. This method is

applied to Propeller DTMB 4119, Propeller NSRDC 4381 and DTMB Duct II

vii

for validation. The results show good correlation with experimental measure-

ments or RANS (ANSYS/FLUENT) results. The method is further used to

develop a viscous image model for the cases of three dimensional wing blades

between two parallel slip walls.

An improved method for hydrofoils and propeller blades with non-zero

thickness or open trailing edges is presented as well. The method in this thesis

follows the idea of Pan (2009, 2011), but applies a new extension scheme, which

uses second order polynomials to describe the extension edges. A improved

simplified search scheme is also used to find the correct shape of the extension

automatically to ensure the two conditions are satisfied.

viii

Table of Contents

Acknowledgments v

Abstract vii

List of Tables xii

List of Figures xiii

Nomenclature xxii

Chapter 1. Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 2. Two Dimensional Viscous/Inviscid Interactive Method 5

2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 2D Wall Transpiration Model . . . . . . . . . . . . . . . . . . . 7

2.3.1 Boundary Layer Simulation . . . . . . . . . . . . . . . . 7

2.3.2 Edge Velocity Expression . . . . . . . . . . . . . . . . . 8

2.4 The Viscous/Inviscid Flow Coupling . . . . . . . . . . . . . . . 14

2.4.1 2D Integral Boundary Layer Equations . . . . . . . . . . 14

2.4.2 Coupling Algorithm . . . . . . . . . . . . . . . . . . . . 15

2.5 2D Viscous Hydrofoil Case . . . . . . . . . . . . . . . . . . . . 15

2.5.1 Method Validation . . . . . . . . . . . . . . . . . . . . . 17

2.5.2 Result Comparison with RANS (ANSYS/FLUENT) . . 19

ix

Chapter 3. Three Dimensional Viscous/Inviscid Interactive Method 28

3.1 Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 3D Wall Transpiration Model . . . . . . . . . . . . . . . . . . . 35

3.2.1 One Blade . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.2 Multiple Blades . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Propeller DTMB4119 . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Propeller NSRDC4381 . . . . . . . . . . . . . . . . . . . . . . 48

3.5 DTMB Duct II . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter 4. Viscous Image Model 68

4.1 Image Model with Panel Method Only . . . . . . . . . . . . . 68

4.2 Image Model with 3D Viscous/Inviscid Interactive Method . . 71

4.3 Straight Wing between Two Parallel Slip Walls . . . . . . . . . 72

4.4 Swept Wing between Two Parallel Slip Walls . . . . . . . . . . 81

Chapter 5. Hydrofoils and Propellers with Non-Zero TrailingEdge Thickness 86

5.1 2D Hydrofoils with Non-Zero Trailing Edge Thickness . . . . . 86

5.1.1 Open Trailing Edge Extension Scheme . . . . . . . . . . 86

5.1.2 Search Scheme for Extension . . . . . . . . . . . . . . . 89

5.2 Sample Cases of 2D Hydrofoils . . . . . . . . . . . . . . . . . 91

5.2.1 Strip of ONR-AxWj-2 Rotor Blade . . . . . . . . . . . . 91

5.2.2 NACA Hydrofoil . . . . . . . . . . . . . . . . . . . . . . 95

5.3 Propeller Blades with Non-zero Trailing Edge Thickness . . . . 98

Chapter 6. Conclusions and Recommendations 107

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 109

Appendix 111

Appendix 1. Evaluation of Influence Coefficients 112

1.1 2D Influence Coefficients . . . . . . . . . . . . . . . . . . . . . 112

1.2 3D Influence Coefficients . . . . . . . . . . . . . . . . . . . . . 112

x

Bibliography 113

Vita 118

xi

List of Tables

2.1 Three grids used for the grid dependency study of the RANS(ANSYS/FLUENT) case for 2D hydrofoil . . . . . . . . . . . . 22

2.2 Information of the RANS (ANSYS/FLUENT) case for 2D hy-drofoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Information of the RANS (ANSYS/FLUENT) case for Pro-peller NSRDC4381 . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1 Information of the RANS (ANSYS/FLUENT) case for a s-traight wing blade between two parallel slip walls. . . . . . . . 81

xii

List of Figures

2.1 Velocity profiles for the real viscous and equivalent inviscid flow 8

2.2 The diagram of the paneling on a 2D hydrofoil and its wake . 9

2.3 Flowchart of the 2D viscous/inviscid boundary layer method . 16

2.4 Pressure distributions along the hydrofoil subject to a unifor-m inflow with zero angle of attack, predicted by the presentmethod with increasing number of panels, Re = U∞c

υ= 5, 000, 000,

fixed transition points at 0.1 chord length on both pressure andsuction sides, 1% turbulence level. . . . . . . . . . . . . . . . . 18

2.5 Diagram of the original hydrofoil and the reversed one (tmax/c =0.1, fmax/c = 0.02). . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Pressure distributions along the two reversed hydrofoils subjectto a uniform inflow with zero angle of attack, predicted by thepresent method, Re = U∞c

υ= 5, 000, 000, fixed transition points

at 0.1 chord length on both the pressure and suction sides, 1%turbulence level. . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Grid details near the leading and trailing edge of the hydrofoilused in RANS (ANSYS/FLUENT) simulations. . . . . . . . . 20

2.8 Pressure coefficients along the hydrofoil subject to a unifor-m inflow with zero angle of attack, predicted by RANS (AN-SYS/FLUENT) using different grids, Re = U∞c

υ= 5, 000, 000. . 21

2.9 y+ along the hydrofoil body subject to a uniform inflow withzero angle of attack, predicted by RANS (ANSYS/FLUENT)using the refined grid a and b, Re = U∞c

υ= 5, 000, 000. . . . . . 22

2.10 Velocity vectors close to the hydrofoil trailing edge in a unifor-m inflow with zero angle of attack, predicted by RANS (AN-SYS/FLUENT), Re = U∞c

υ= 5, 000, 000. . . . . . . . . . . . . 24

2.11 Diagram of the eight selected eight points . . . . . . . . . . . 25

2.12 Velocity profiles at point 2 and 4 on the hydrofoil surface in auniform inflow with zero angle of attack, predicted by RANS(ANSYS/FLUENT), Re = U∞c

υ= 5, 000, 000. . . . . . . . . . . 25

xiii

2.13 Comparison of δ∗ on the surface of the hydrofoil subject to auniform inflow with zero angle of attack, between RANS (AN-SYS/FLUENT) and the present method, Re = U∞c

υ= 5, 000, 000. 26

2.14 Comparison of Cp on the surface of the hydrofoil subject to auniform inflow with zero angle of attack, between RANS (AN-SYS/FLUENT) and the present method, Re = U∞c

υ= 5, 000, 000. 26

2.15 Comparison of Cf on the surface of the hydrofoil subject to auniform inflow with zero angle of attack, between RANS (AN-SYS/FLUENT) and the present method, Re = U∞c

υ= 5, 000, 000. 27

3.1 Blade geometry of Propeller NSRDC4381 . . . . . . . . . . . . 30

3.2 Velocity components in t and s directions at the middle chordof the strip r/R = 0.6, of propeller 4381, predicted by RANS(ANSYS/FLUENT), Js = 0.889. . . . . . . . . . . . . . . . . . 31

3.3 Velocity components in t and s directions at the end chord(close to trailing edge) of the strip r/R = 0.6, of propeller 4381,predicted by RANS (ANSYS/FLUENT), Js = 0.889. . . . . . 31

3.4 Velocity components in t and s directions at the middle chordof the strip r/R = 0.95, of propeller 4381, predicted by RANS(ANSYS/FLUENT), Js = 0.889. . . . . . . . . . . . . . . . . . 32

3.5 Velocity components in t and s directions at the end chord(close to trailing edge) of the strip r/R = 0.95, of propeller4381, predicted by RANS (ANSYS/FLUENT), Js = 0.889. . . 32

3.6 Velocity components in t and s directions at the middle chordof the strip r/R = 0.6, of propeller 4381, predicted by RANS(ANSYS/FLUENT), Js = 0.5. . . . . . . . . . . . . . . . . . . 33

3.7 Velocity components in t and s directions at the end chord(close to trailing edge) of the strip r/R = 0.6, of propeller 4381,predicted by RANS (ANSYS/FLUENT), Js = 0.5. . . . . . . . 33

3.8 Velocity components in t and s directions at the middle chordof the strip r/R = 0.95, of propeller 4381, predicted by RANS(ANSYS/FLUENT), Js = 0.5. . . . . . . . . . . . . . . . . . . 34

3.9 Velocity components in t and s directions at the end chord(close to trailing edge) of the strip r/R = 0.95, of propeller4381, predicted by RANS (ANSYS/FLUENT), Js = 0.5. . . . 34

3.10 Sketch of 3D blade paneling . . . . . . . . . . . . . . . . . . . 36

3.11 Flowchart of the iterative scheme for 3D viscous/inviscid bound-ary layer method . . . . . . . . . . . . . . . . . . . . . . . . . 38

xiv

3.12 Pressure distributions along two different strips, r/R=0.6 andr/R=0.9, of Propeller NSTDC 4381, predicted by the presentmethod with and without the effects of blowing sources addedon neighboring blades, Js = 0.889, Re = U∞D

υ= 7.42e5, 1%

turbulence level, fixed transition points at 0.1 chord length onboth the pressure and suction sides of each strip. . . . . . . . 40

3.13 Paneled geometry of Propeller DTMB4119 . . . . . . . . . . . 41

3.14 Pressure distributions predicted by the present method usingdifferent numbers of panels along the strip r/R = 0.62 of Pro-peller DTMB4119, Js = 0.833, Re = U∞D

υ= 766, 395, 1% tur-

bulence level, fixed transition points at 0.1 chord length on boththe pressure and suction sides of each strip. . . . . . . . . . . 42

3.15 Pressure distributions at different iterations on the strip r/R =0.62 of Propeller DTMB4119, predicted by the present method,Js = 0.833, Re = U∞D

υ= 766, 395, 1% turbulence level, fixed

transition points at 0.1 chord length on both the pressure andsuction sides of each strip. . . . . . . . . . . . . . . . . . . . . 43

3.16 Comparison of displacement thickness on the pressure side ofthe strip r/R = 0.7 of Propeller DTMB4119 between exper-iments and the present method (free transition point on thepressure side of each strip, 1% turbulence level), Js = 0.833,Re = U∞D

υ= 766, 395. . . . . . . . . . . . . . . . . . . . . . . . 44

3.17 Comparison of displacement thickness on the suction side of thestrip r/R = 0.7 of Propeller DTMB4119 between experimentsand the present method (fixed transition point at 0.5 chordlength on the suction side of each strip, 1% turbulence level),Js = 0.833, Re = U∞D

υ= 766, 395. . . . . . . . . . . . . . . . . 44

3.18 Comparison of pressure coefficients on the strip, r/R = 0.3 of,of Propeller DTMB4119 between experiments and the presentmethod (fixed transition point at 0.5 chord length on the suctionside and free transition point on the pressure side of each strip,1% turbulence level), Js = 0.833, Re = U∞D

υ= 766, 395. . . . . 45


υ= 766, 395. . . . . 46

xv


υ= 766, 395. . . . . 47

3.21 Comparisons of KT and KQ of Propeller DTMB4119 at differentadvance ratios, between experiments and the present method. 49

3.22 Paneled geometry of Propeller NSRDC4381. . . . . . . . . . . 50

3.23 Domain and boundary conditions of the RANS (ANSYS/FLUENT)case for Propeller NSRDC4381 (from Sharma 2011). . . . . . . 51

3.24 Grid details used in the RANS (ANSYS/FLUENT) case forPropeller NSRDC4381. Top left: O type grid on the propellerblade. Top right: grid details on the hub around the root sectionof the propeller blade. Bottom left: Grid details on the hub nearthe leading edge of the root section of propeller blade. Bottomright: grid details on the hub near the trailing edge of the rootsection of the propeller blade. (from Sharma 2011). . . . . . . 52

3.25 Pressure distributions predicted by the present method usingdifferent numbers of panels at strips, r/R = 0.6 and r/R = 0.9,of Propeller NSRDC4381, Js = 0.889, Re = U∞D

υ= 7.42e5, 1%

turbulence level, fixed transition points at 0.1 chord length onboth the pressure and suction sides of each strip. . . . . . . . 53

3.26 Pressure distributions at different iterations at strips r/R = 0.6and r/R = 0.9 of Propeller NSRDC4381, predicted by thepresent method, Js = 0.889, Re = U∞D

υ= 7.42e5, 1% turbu-

lence level, fixed transition points at 0.1 chord length on boththe pressure and suction sides of each strip. . . . . . . . . . . 54

3.27 Comparison of pressure distributions along the strip, r/R = 0.6,of Propeller NSRDC4381, predicted by RANS (ANSYS/FLUENT),panel method and the present method (fixed transition points at0.1 chord length on both the pressure and suction sides of eachstrip, 1% turbulence level), Js = 0.889, Re = U∞D

υ= 7.42e5 for

viscous cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.28 Comparison of pressure distributions along the strip, r/R = 0.8,of Propeller NSRDC4381, predicted by RANS (ANSYS/FLUENT),panel method and the present method (fixed transition points at0.1 chord length on both the pressure and suction sides of eachstrip, 1% turbulence level), Js = 0.889, Re = U∞D

υ= 7.42e5 for

viscous cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

xvi

3.29 Comparison of pressure distributions along the strip, r/R = 0.6,of Propeller NSRDC4381, predicted by RANS (ANSYS/FLUENT),panel method and the present method (fixed transition pointsat 0.1 chord length on both the pressure and suction sides ofeach strip, 1% turbulence level), Js = 0.5, Re = U∞D

υ= 4.15e5

for viscous cases. . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.30 Comparison of pressure distributions along the strip, r/R = 0.8,of Propeller NSRDC4381, predicted by RANS (ANSYS/FLUENT),panel method and the present method (fixed transition pointsat 0.1 chord length on both the pressure and suction sides ofeach strip, 1% turbulence level), Js = 0.5, Re = U∞D

υ= 4.15e5

for viscous cases. . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.31 Comparison of Cf at strips, r/R = 0.6 and r/R = 0.6, of Pro-peller NSRDC4381, between RANS (ANSYS/FLUENT) andthe present method (fixed transition points at 0.1 chord lengthon both the pressure and suction sides of each strip, 1% turbu-lence level), Js = 0.889, Re = U∞D

υ= 7.42e5. . . . . . . . . . . 59

3.32 Comparisons of KT and KQ of Propeller NSRDC4381 at differ-ent advance ratios, between the present method combined withPSF2 wake alignment and experiments. . . . . . . . . . . . . . 60

3.33 Comparisons of KT and KQ of Propeller NSRDC4381 at differ-ent advance ratios, between the present method combined withfull wake alignment and experiments. . . . . . . . . . . . . . . 61

3.34 Paneled geometry of the DTMB Duct II. . . . . . . . . . . . . 62

3.35 Pressure distributions on DTMB Duct II predicted by the presentmethod using different numbers of panels on both the chordand span wise directions, Re = U∞D

υ= 2.06e6, fixed transition

points at 0.05 chord length on both the pressure and suctionsides of each strip, 1% turbulence level. . . . . . . . . . . . . . 63

3.36 Pressure distributions at different iterations on DTMB Duct II,predicted by the present method, Re = U∞D

υ= 2.06e6, fixed

transition points at 0.05 chord length on both the pressure andsuction sides of each strip, 1% turbulence level. . . . . . . . . 64

3.37 Pressure distributions along three different sections of DTMBDuct II, predicted by the present method, Re = U∞D

υ= 2.06e6,

fixed transition points at 0.05 chord length on both the pressureand suction sides of each strip, 1% turbulence level. . . . . . . 65

xvii

3.38 Displacement thickness of boundary layer along three differen-t sections of DTMB Duct II, predicted by the present method,Re = U∞D

υ= 2.06e6, fixed transition points at 0.05 chord length

on both the pressure and suction sides of each strip, 1% turbu-lence level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.39 Viscous and inviscid pressure distributions on the DTMB DuctII predicted by the present method (fixed transition points at0.05 chord length on both the pressure and suction sides of eachstrip, 1% turbulence level), compared with the experimentalmeasurements by Morgan and Caster (1965), Re = U∞D

υ= 2.06e6. 67

4.1 Diagram of a straight wing blade adjacent to a slip wall . . . 69

4.2 Diagram of a straight wing blade and its images, symmetricabout the red line . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Diagram of the equivalence of the influence coefficients, fromSingh (2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Diagram of a wing blade between two parallel walls . . . . . . 72

4.5 Pressure distributions along the strip at r/R = 0.75, predictedby the present method using different numbers of panels on spanand chord wise directions, Re = U∞c


points at 0.1 chord length on both the pressure and suction sidesof each strip, 1% turbulence level. . . . . . . . . . . . . . . . 73

4.6 Circulations on the straight wing blade predicted by the originalimage model of Singh (2009) using different numbers of images. 74

4.7 Pressure distributions along the strip at r/R = 0.6 of the s-traight wing blase, predicted by the original and the presentimage model, using different numbers of images. For the vis-cous case, Re = U∞c

υ= 1.0e6, fixed transition points at 0.1 chord

length on both the pressure and suction sides of each strip, 1%turbulence level. . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.8 Pressure distributions on the strips at r/R = 0.3, r/R = 0.5 andr/R = 0.9 of the straight wing blade, predicted by the originaland the present image model, using different numbers of images.For the viscous case, Re = U∞c

υ= 1.0e6, fixed transition points

at 0.1 chord length on both the pressure and suction sides ofeach strip, 1% turbulence level. . . . . . . . . . . . . . . . . . 76

4.9 Diagram of the domain of the straight wing blade between twoparallel slip walls for 3D RANS (ANSYS/FLUENT) simulations. 77

4.10 Grid details around the straight wing blade used in 3D RANS(ANSYS/FLUENT) simulations. . . . . . . . . . . . . . . . . 78

xviii

4.11 Pressure coefficients on the strips at r/R=0.2 and r/R=0.6 ofthe 3D straight wing predicted by the 3D RANS (ANSYS/FLUENT)case, and on the 2D strip predicted by the 2D RANS (AN-SYS/FLUENT) case, Re = U∞c

υ= 1.0e6. . . . . . . . . . . . . 79

4.12 Pressure coefficients along the strip at r/R=0.5 of the 3D s-traight wing predicted by RANS (ANSYS/FLUENT), the presentmethod (fixed transition points at 0.1 chord length on both thepressure and suction sides of each strip, 1% turbulence level)and the panel method. For the viscous case, Re = U∞c

υ= 1.0e6. 80

4.13 Diagram of a swept wing between two parallel slip walls. . . . 82

4.14 Pressure distributions on the strips at r/R = 0.3, r/R = 0.5 andr/R = 0.9 of the wing blade, predicted by the present method,Re = U∞c

υ= 1.0e6, fixed transition points at 0.1 chord length on

both the pressure and suction sides of each strip, 1% turbulencelevel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.15 Pressure coefficients on the strip at r/R=0.025 of the 3D sweptwing, predicted by the 3D RANS (ANSYS/FLUENT) case, thepresent method (fixed transition points at 0.1 chord length onboth the pressure and suction sides of each strip, 1% turbulencelevel) and the panel method. For the viscous case, Re = U∞c

υ=

1.0e6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


υ=

1.0e6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84


υ=

1.0e6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.1 Section with a non-zero thickness trailing edge, extracted fromthe ONR-AxWj-2 rotor blade . . . . . . . . . . . . . . . . . . 87

5.2 Diagram of the modification of the trailing edge with non-zerothickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Diagram of the extension after the open trailing edge, from Pan(2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xix

5.4 Diagram of extension of the trailing edge with non-zero thickness 90

5.5 Diagram of the procedure of the search scheme for extension . 91

5.6 Comparison of pressure distributions on the strip extracted fromthe ONR-AxWJ-2 rotor blade at r/R = 0.7, between RANS(ANSYS/FLUENT) and the present method (fixed transitionpoints at 0.01 chord length on both the pressure and suctionsides, 1% turbulence level), Re = U∞c

υ= 5.0e6. . . . . . . . . 92

5.7 Final extension behind the the strip extracted from the ONR-AxWJ-2 rotor blade at r/R = 0.7, predicted by the presentmethod, Re = U∞c


length on both the pressure and suction sides, 1% turbulencelevel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.8 Pressure difference ∆p of the cut points vs. vertical position ofthe last camber point yC , of the ONR-AxWJ-2 rotor blade atr/R = 0.7, from the present method, Re = U∞c

υ= 5.0e6, fixed

transition points at 0.01 chord length on both the pressure andsuction sides, 1% turbulence level. . . . . . . . . . . . . . . . . 94

5.9 Comparison of pressure distributions along the NACA hydrofoilwith a non-zero thickness trailing edge, between RANS (AN-SYS/FLUENT) and the present method (fixed transition pointsat 0.01 chord length on both the pressure and suction sides, 1%turbulence level), Re = U∞c

υ= 9.0e6. . . . . . . . . . . . . . . 95

5.10 Final extension behind the NACA hydrofoil with a non-zerothickness trailing edge, predicted by the present method, Re =U∞cυ

= 5.0e6, fixed transition points at 0.01 chord length onboth the pressure and suction sides, 1% turbulence level. . . . 96

5.11 Pressure difference ∆p of the cut points vs. vertical position ofthe last camber point yC , of the NACA hydrofoil with a non-zerothickness trailing edge, from the present method, Re = U∞c

υ=

5.0e6, fixed transition points at 0.01 chord length on both thepressure and suction sides, 1% turbulence level. . . . . . . . . 97

5.12 Propeller with non-zero trailing edge thickness, from Pan (2009). 99

5.13 Propeller with closed sharp trailing edge modified by the presentscheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.14 Pressure differences between the cut points of strip 1 and neigh-boring strips vs. the movement of the last camber point of strip1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.15 Pressure differences between the cut points of strip 5 and neigh-boring strips vs. the movement of the last camber point of strip5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

xx

5.16 Pressure differences between the cut points of strip 10 andneighboring strips vs. the movement of the last camber pointof strip 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.17 Pressure differences between the cut points of strip 18 andneighboring strips vs. the movement of the last camber pointof strip 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.18 Pressure differences between the cut points on the initial andfinal extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.19 Circulation on the final blade geometry. . . . . . . . . . . . . . 104

5.20 KT of the propeller with finite thickness trailing edge. . . . . . 105

5.21 KQ of the propeller with finite thickness trailing edge. . . . . . 106

xxi

Nomenclature

Latin Symbols

Cτ shear stress coefficient Cτ = τmax/(ρU2)

Cf skin-friction coefficient Cf = τwall/(0.5ρU2)

Cp pressure coefficient, Cp = (P − Po)/(0.5ρn2D2)

D propeller diameter, D = 2R

D dissipation or shear work integral, D =∫ δ0

(τ1

∂u∂x3

+ τ2∂w∂x3

)dx3

fmax/C maximum camber to chord ratio

G Green’s function

H shape factor, H = δ∗/θ

H∗ kinetic energy shape factor, θ∗/θ

J advance ratio based on Vs, J = Vs/(nD)

KQ torque coefficient, KQ = Q/(ρn2D5)

KT thrust coefficient, KT = T/(ρn2D4)

xxii

q total velocity

Q torque

U in local inflow velocity (in the propeller fixed system)

u Perturbation velocity

qe Magnitude of edge velocity

R propeller radius

Re Reynolds number

SP blade surface

SW wake surface

K thrust

tmax/C maximum thickness to chord ratio

ue boundary layer edge velocity

uτ wall shear velocity, uτ =√τwall/ρ

x, y, z propeller fixed coordinates

y+ non-dimensional wall distance, y+ = uτyν

xxiii

Greek Symbols

α angle of attack

δ∗ displacement thickness, δ∗ =∫

(1− uUe

)dz

∆t time step size

ω propeller angular velocity

ν kinematic viscosity of water

φ perturbation potential

Φ total potential

ρ fluid density

θ momentum thickness, θ =∫

uUe

(1− uUe

)dz

θ∗ kinetic energy thickness, θ∗ =∫

uUe

(1− u2

U2e)dz

Uppercase Abbreviations

2D two dimensional

3D three dimensional

CPU central processing unit

MIT Massachusetts Institute of Technology

NACA National Advisory Committee for Aeronautics

RANS Reynolds-averaged Navier Stokes

xxiv

Computer Program Names

CAV2DBL cavitating 2-dimensional with boundary layer

Fluent A commercial RANS solver

PROPCAV cavitating propeller potential flow solver based on BEM

XFOIL 2D integral boundary layer analysis code

xxv

Chapter 1

Introduction

1.1 Background

Panel methods have been extensively used in the design stage of ma-

rine propellers due to the satisfactory prediction of propeller performance with

small computational cost. However, since they are based on the inviscid po-

tential flow theory, classical panel methods cannot predict the effects of fluid

viscosity. For propeller flows, viscosity can not only cause friction but also af-

fect the pressure on propeller blades, resulting in the changes of thrust/torque

forces and propeller efficiency. Some empirical methods can be used with pan-

el methods to take account of the effects of fluid viscosity. Kerwin and Lee

(1978) used an adopted constant as the skin friction coefficient to include the

contribution of the viscous shear stress and an empirical viscous pitch cor-

rection to approximate the influences of the boundary layer. However, such

empirical methods require user adjustable constants, which could be good for

some cases, but inaccurate for some other cases. In order to handle the effect-

s of viscosity for general blade geometries and various flow conditions, more

rational methods are required.

Viscous/inviscid interactive methods in 2D have been highly developed

1

in the past decades, from which the fully simultaneous coupling scheme devel-

oped by Drela (1985, 1987) has been proved to be the most robust and efficient

one for 2-D separated flows. Drela (1989) implemented a viscous/inviscid in-

teractive solver, XFOIL, for 2D airfoils. XFOIL uses the stream function to

get the inviscid flow field. Then the inviscid solution is coupled with the 2D in-

tegral boundary layer equations to solve for the boundary layer variables. The

development of the viscous/inviscid interactive methods in 3D is limited be-

cause of the complexity of 3D boundary layer. Cousteix and Houdeville (1981)

pointed out that the 3D integral boundary layer equations consist of three fully

hyperbolic non-linear partial differential equations. Milewski (1997) develope-

d a 3D fully simultaneous viscous/inviscid interactive scheme, by coupling a

potential based panel method and the 3D integral boundary layer equation-

s, for 3D wetted hydrofoil and duct flows. However, such a scheme requires

a large amount of computer resources and is very difficult to be applied to

complicated geometries such as propeller blades. For 3D propeller cases par-

ticularly, Jessup (1989) measured the boundary layer variables on the blades

of Propeller DTMB 4119 and found that the boundary layer variables on the

stream wise direction are much more significant than those on the span wise

direction. According to Jessup’s measurements, Hufford (1992, 1994) assumed

that the effects of the boundary layer on propeller blades should be negligible

radially outward compared with those along constant radius. Based on the

above assumption, he proposed a method used to predict the 3D boundary

layer effects on propeller blades. His method calculates the inviscid solution

2

by a potential based panel method and then applies a 2D viscous/inviscid

interactive method to each strip along constant radius of the blades using,

however, 3D boundary layer sources over each strip. Sun (2008) coupled a

potential based panel method code, PROPCAV, with XFOIL following Huf-

ford’s method, but further simplified the method by adding 2D boundary layer

sources over the strips. However, Hufford and Sun considered the boundary

layer on each strip independently and ignored the boundary layer effects of

neighboring strips and other blades, which oversimplified the problem.

1.2 Objective

The main objective of this thesis is to develop a more reasonable and

complete 3D viscous/inviscid interactive method by coupling a 3D low-order

potential based panel method code, PROPCAV, with Drela’s 2D integral

boundary layer solver, XFOIL, to investigate the effects of fluid viscosity on

the performance of propellers.

1.3 Organization

This thesis is organized into six chapters.

Chapter 1 contains the literature review, the objective and the organi-

zation of this study.

Chapter 2 introduces a two-dimensional viscous/inviscid interactive

method in details. This method is applied to a 2D hydrofoil case, and the

3

results are correlated with RANS (ANSYS/FLUENT) results.

Chapter 3 develops a three dimensional viscous/inviscid interactive

method for propeller and duct flows. The method follows the assumption

of Hufford (1992, 1994) and Sun (2008) that the boundary layer along the

span wise direction (radially outward for propeller blades) can be negligible,

but further considers the boundary layer effects from neighboring strips and

blades. Then this method is validated through two propeller and a bare duct

cases.

Chapter 4 extends the three dimensional viscous/inviscid interactive

method to an image model used to consider the effects of slip walls. Cases of

a straight and a swept wing blade between two parallel walls are carried out

to validate the image model.

Chapter 5 improves the scheme proposed by Pan (2009, 2011) for hydro-

foils and propeller blades with non-zero thickness trailing edges. The results

of sample cases are presented.

Chapter 6 summarizes the work of this thesis and presents the recom-

mendations for future research.

4

Chapter 2

Two Dimensional Viscous/Inviscid Interactive

Method

In this chapter, a 2D viscous/inviscid interactive method is introduced.

This method couples a 2D low-order potential based panel method and the 2D

integral boundary layer analysis. The chapter starts with the introduction of

the 2D wall transpiration model and then derives the expression of the viscous

or edge velocity. In the following, the 2D integral boundary layer equations are

summarized. The coupling algorithm between the invsicid and viscous solution

is presented next. Finally, this method is applied to a 2D hydrofoil case. The

results are compared with those predicted by RANS (ANSYS/FLUENT).

2.1 Governing Equations

A lower-order panel method, in which constant sources and dipoles

are used, based on perturbation potential is applied in this thesis. The to-

tal velocity is decomposed into two components: the inflow velocity and the

perturbation velocity.

q = U in + u (2.1)

5

where q is the total velocity, Uin is the velocity of incoming flow, and u is the

perturbation velocity.

u = ∇φ (2.2)

q = ∇Φtotal (2.3)

The perturbation potential φ satisfies the Laplace Equation:

∇2φ = 0 (2.4)

The Laplace Equation can be integrated on the boundaries:

φp2

=

∫∫Sp

[−φp′

∂G(p, p′)

∂np′+∂φp′

∂np′G(p, p′)

]dS

−∫∫

SW

∆φw(p′)∂G(p, p′)

∂np′dS

(2.5)

where p and p′ correspond to the field point and the variable point; G(p, p′) =

− 14πr

for three dimension or G(p, p′) = ln(r)2π

for two dimension, is the Green’s

function with r being the distance between the field point p and the variable

point p′; ∆φw is the potential jump across the wake; SP and SW represent the

blade surface and wake sheet, respectively.

2.2 Boundary Conditions

In order to obtain an unique solution of (2.5), appropriate boundary

conditions are applied to the flow domain. For fully wetted cases, the kinematic

boundary condition states:

∂φ

∂n= −U in · n (2.6)

6

where n is the normal vector on the blade surface pointing into the flow field.

At the trailing edge of the blade, Kutta condition is applied, which implies the

finite velocity at the blade trailing edge.

∇φ = finite (2.7)

At the far field, the perturbation velocity vanishes.

∇φ = 0 (2.8)

2.3 2D Wall Transpiration Model

A 2D wall transpiration model is introduced in this section to simulate

the 2D viscous boundary layer. This model couples the potential and the

2D integral boundary layer equations through the edge velocity. The detailed

derivation of the expression of the edge or viscous velocity is presented.

2.3.1 Boundary Layer Simulation

For the potential flow, blowing sources are added on the wall to simulate

the boundary layer in the real viscous flow (shown in Fig. 2.1). The blowing

sources should be given proper strengthes so that the velocity at the edge

of boundary layer, y = ye, is the same with that in the real viscous flow.

According to the Bernoulli equation, the same edge velocities in the two flows

lead to the same pressures on y = ye. In addition, the boundary layer theory

tells that the pressure cross the boundary layer stays constant, which means

that the pressures on the point A of the edge of boundary layer and point B

7

of the wall are the same. Thus it is concluded that the pressure on the wall

in the equivalent inviscid flow is the same with that in the real viscous flow.

So far, the effects of the boundary layer on the pressure have been considered

in the equivalent inviscid flow by adding proper blowing sources on the wall.

In addition, the friction on the wall can be considered by solving the integral

boundary layer equations.

𝑦 = 𝑦𝑒

A

B

(a) Real viscous flow

𝑦 = 𝑦𝑒

Induced normal velocity by the sources

A’

(b) Equivalent inviscid flow

Figure 2.1: Velocity profiles for the real viscous and equivalent inviscid flow

2.3.2 Edge Velocity Expression

The expression of the edge velocity Ue or viscous velocity Uvis for 2D

hydrofoil cases are derived in this subsection. For 2D hydrofoil cases (shown

in Fig. 2.2), (2.5) becomes:

πφp =

∫Sp

(∂φ

∂nln r − φ∂ ln r

∂n

)ds−

∫SW

(∆φw

∂ ln r

∂n+

)ds (2.9)

8

where φp is the perturbation potential on the hydrofoil body. The discretized

expression of (2.9) is

N∑j=1

Dijφj =N∑j=1

Sij

(∂φ

∂n

)j

−Wi∆φw (2.10)

The Kutta condition can be satisfied using a modified Morino Kutta condition

proposed by Kinnas et al. (1990)

∆φ = ΦN − Φ1 + U in · rt (2.11)

where rt is the distance between the control points of the upper and lower

trailing edge panels (the 1st and the N th panel). Then (2.10) becomes:

N∑j=1

Aijφj =N∑j=1

Sij

(∂φ

∂n

)j

−Wi (U in · rt) (2.12)

(2.12) can be used to calculate the perturbation potential on the hydrofoil. To

consider the effects of fluid viscosity, blowing sources with unknown strength

are added on both the body and the wake panels to simulate viscous boundary

layer. When the added blowing sources are taken into consideration, (2.12)

𝑼𝒊𝒏

n

𝒏+

Figure 2.2: The diagram of the paneling on a 2D hydrofoil and its wake

9

becomes:

N∑j=1

Aijφj =N∑j=1

Sij

(∂φ

∂n

)j

−Wi (U in · rt) +

N+NW∑j=1

Bijσj (2.13)

where σj is the unknown strength of the blowing source added at the panel

j; Bij is the source induced influence coefficient analogous to Sij; N and NW

indicate the number of panels on the hydrofoil body and the wake, respectively.

Then the perturbation potential on the body panel can be obtained from

(2.13):

φi =N∑j=1

A−1ij

[N∑j=1

Sjk

(∂φ

∂n

)k

−Wi (U in · rt)

]

+

N+NW∑j=1

(N∑j=1

A−1ik Bkj

)σj

(2.14)

(2.14) is the expression of the perturbation potential taken the added blowing

sources into consideration, or it can be called the viscous perturbation poten-

tial. The total potential for viscous cases should be the sum of the viscous

perturbation potential and the inflow potential:

Φi,vis = Φini + φi,vis

= Φini +

N∑j=1

A−1ij

[N∑j=1

Sjk

(∂φ

∂n

)k

−Wi (U in · rin)

]+

N+NW∑j=1

(N∑j=1

A−1ij Bkj

)σj

= Φini + φi,inv +

N+NW∑j=1

Hijσj

= Φi,inv +

N+NW∑j=1

Hijσj

(2.15)

10

where Hij =∑N

j=1A−1ij Bkj. According to (2.15), the total viscous potential

Φi,ivs consists of the total inviscid potential Φi,inv and the induced potential

by the added blowing sources∑N+NW

j=1 Hijσj. The viscous velocity or edge

velocity can be obtained by differentiating the total viscous potential along

the surface of the hydrofoil:

Ui,vis =∂Φi,ivs

∂s

=∂Φi,inv

∂s+

∂

∂s

(N+NW∑j=1

Hijσj

)

= Ui,inv +

N+NW∑j=1

Cijσj

(2.16)

The viscous velocity on the hydrofoil body can be expressed as (2.16). The

derivative of Hij can be achieved by:

Cij =∂Hij

∂s=Hi,j −Hi−1,j

∆s(2.17)

(2.16) states that the viscous velocity Ui,vis is the sum of the inviscid velocity

Ui,inv and the induced velocity by the added blowing sources∑N+NW

j=1 Cijσj.

It should be noted that the inflow potential Φin does not exist in the cases

with rotational inflows. Even though the derivation of (2.16) uses the inflow

potential Φin, (2.16) is still general and valid for cases with or without the

inflow potential. This is because (2.16) is an expression in terms of velocity

instead of potential.

The viscous velocity on the wake sheet can be derived similarly. With

the added blowing sources, the perturbation potential (viscous perturbation

11

potential) on the wake can be written as

φwake =1

2π

∫SP

(∂φ

∂nln r − φ∂ ln r

∂n

)ds+

1

2π

∫SP+SW

(ln r)σds (2.18)

Discretizing (2.18) gives:

φwakei =N∑j=1

Awijφj +N∑j=1

Cwij

(∂φ

∂n

)j

+

N+NW∑j=1

Bwijσj (2.19)

where Awij, Cwij and Bw

ij are the influence coefficients analogues to those in (2.13)

for hydrofoil body. φj in (2.19) is the viscous perturbation potential on the

panel j of the hydrofoil body, which is the sum of the inviscid perturbation

potential and the induced potential due to the blowing sources:

φj = φj,inv +

N+NW∑j=1

Hjkσk (2.20)

Replace φj in (2.19) with (2.20) and gives:

φwakei =N∑j=1

Awij

[φinvj +

N+NW∑j=1

Hjkσk

]+

N∑j=1

Cwij

(∂φ

∂n

)j

+

N+NW∑j=1

Bwijσj

=N∑j=1

Awijφj,inv +N∑j=1

Cwij

(∂φ

∂n

)j

+

N+NW∑j=1

(N∑k=1

AwjkHkj +Bwij

)σj

= φwakei,inv +

N+NW∑j=1

Hwijσj

(2.21)

The viscous total potential should be:

Φwakei,vis = Φin

i + φwakei,vis

= Φini + φwakei,inv +

N+NW∑j=1

Hwijσj

= Φwakei,inv +

N+NW∑j=1

Hwijσj

(2.22)

12

Then the viscous or edge velocity can be written as:

Uwakei,ivs =

∂Φwakei,vis

∂s

=∂Φwake

i,inv

∂s+

∂

∂s

(N+NW∑j=1

Hwijσj

)

= Uwakei,inv +

N+NW∑j=1

Cwijσj

(2.23)

(2.16) and (2.23) express the viscous velocity on the hydrofoil body and wake

using the inviscid velocity, which can be solved by the low-order potential

based panel method, and the induced velocity by the added blowing sources.

The viscous velocity on the body and wake can be written in a general form:

Ui,vis = Ui,inv +

N+NW∑j=1

Gijσj (2.24)

where Gij = Cij on the hydrofoil body and Gij = Cwij on the wake. i is from

1 to N +NW . The unknown strength of the blowing source is directly related

to the boundary layer values by:

σ =∂ (Uvisδ

∗)

∂s=∂m

∂s(2.25)

where δ∗ is the displacement thickness of boundary layer, and m = Uvisδ∗ is

defined as the mass defect. (2.25) can be calculated numerically

σj =mj+1 −mj

∆s(2.26)

By replacing the blowing source strength σj in (2.24) with (2.26), the final

expression for viscous velocity is obtained:

Ui,vis = Ui,inv +

N+NW∑j=1

Dijmj (2.27)

13

where

Di,j =Gi,j−1

sj − sj−1− Gi,j

sj+1 − sj(2.28)

2.4 The Viscous/Inviscid Flow Coupling

(2.27) needs to be combined with the 2D integral boundary layer equa-

tions. Then the combined equations can be solved together.

2.4.1 2D Integral Boundary Layer Equations

The 2D integral boundary layer equations include the standard integral

momentum and the kinetic energy equation. A closure equation is used to close

the system. For incompressible flow, the 2D integral boundary layer equations

can be expressed as the followings:

• Momentum equation

∂θ

∂s+ (2 +H)

θ

Ue

dUeds

=Cf2

(2.29)

• Kinetic energy equation

θdH∗

ds+H∗(1−H)

θ

Ue

dUeds

= 2CD −H∗Cf2

(2.30)

14

• Closure

Closure for turbulent flows

δ

Cτ

dCτds

= 5.6[C

12τEQ − C

12τ

]+ 2δ

{4

3δ∗

[Cf2−(Hk − 1

6.7Hk

)2]− 1

Ue

dUeds

} (2.31)

Closure for laminar flows

dn

ds=dn(Hk)

dReθ

Reθ(Hk, θ)

ds(2.32)

2.4.2 Coupling Algorithm

The brief solution procedures of this viscous/inviscid interactive method

includes the following steps (as shown in Fig. 2.3): First, the inviscid velocity

Uinv is calculated by the low-order potential based panel method. Then the

viscous or the edge velocity Uvis is given as the inviscid velocity. The 2D in-

tegral boundary layer equations and the closure can be solved to obtain the

boundary layer variables. Next, the viscous velocity is updated using (2.24).

Repeat this process until a converged result is obtained.

2.5 2D Viscous Hydrofoil Case

A 2D hydrofoil case is carried out in order to validate the 2D vis-

cous/inviscid interactive method and test how this method can capture the

boundary layer characteristics. The hydrofoil is generated based on the NACA00

thickness form with tmax/c = 0.1. A camber with fmax/c = 0.02 is added. It

15

The inviscid velocity 𝑈𝑖𝑛𝑣

(low-order potential based panel method)

Viscous or edge velocity 𝑈𝑒

Solve integral BL equations

Initially 𝑈𝑒 = 𝑈𝑖𝑛𝑣

Update 𝑈𝑒

Converged?

No

Yes Final results

Figure 2.3: Flowchart of the 2D viscous/inviscid boundary layer method

16

is subject to a uniform inflow with zero angle of attack. The transition points

on both pressure and suction side of the hydrofoil used in the present method

are fixed close to the leading edge. The turbulence level is given as 1%. The

Reynolds number is selected to be 5,000,000. The results predicted by the

2D viscous/inviscid interactive method and RANS (ANSYS/FLUENT) are

compared.

2.5.1 Method Validation

A grid dependency study of the present viscous/inviscid interactive

method is conducted first. The pressure distributions along the hydrofoil pre-

dicted by the present method using different numbers of panels are plotted in

Fig. 2.4. The plots in Fig. 2.4 collapse into the same line, indicating that the

changes of the number of panels on the hydrofoil do not affect the final results.

Another case of a reversed hydrofoil is conducted to validate the present

method. The same NACA hydrofoil is reversed up and down as shown in Fig.

2.5. Subject to the same inflow, the pressure on the reversed hydrofoil is

expected the same with that on the original hydrofoil. Fig. 2.6 plots the

pressure distributions on the two hydrofoils predicted by the present method.

The practically identical curves shown in Fig. 2.6 provide a good verification

test of the present method.

17

Figure 2.4: Pressure distributions along the hydrofoil subject to a uniform in-flow with zero angle of attack, predicted by the present method with increasingnumber of panels, Re = U∞c

υ= 5, 000, 000, fixed transition points at 0.1 chord

length on both pressure and suction sides, 1% turbulence level.

18

2.5.2 Result Comparison with RANS (ANSYS/FLUENT)

A grid dependence study of the RANS case is conducted first. The

RANS simulations are carried out in a rectangular domain. A free stream

condition with a specified velocity is imposed on the left, top and bottom

boundaries. The outflow condition is set at the right boundary as the outlet

of the domain. The grid details around the leading and trailing edge of the

hydrofoil are shown in Fig. 2.7. The grid is refined and the three grids used

in the grid dependency study are listed in Table 2.1. The plots of the pressure

coefficient along the hydrofoil predicted by RANS using these three grids col-

lapse into a same line (shown in Fig. 2.8), indicating that the grids work fine

for this case.

Figure 2.5: Diagram of the original hydrofoil and the reversed one (tmax/c =0.1, fmax/c = 0.02).

19

Figure 2.6: Pressure distributions along the two reversed hydrofoils subject toa uniform inflow with zero angle of attack, predicted by the present method,Re = U∞c

υ= 5, 000, 000, fixed transition points at 0.1 chord length on both the

pressure and suction sides, 1% turbulence level.

(a) Leading edge (b) Trailing edge

Figure 2.7: Grid details near the leading and trailing edge of the hydrofoilused in RANS (ANSYS/FLUENT) simulations.

20

Figure 2.8: Pressure coefficients along the hydrofoil subject to a uniform in-flow with zero angle of attack, predicted by RANS (ANSYS/FLUENT) usingdifferent grids, Re = U∞c

υ= 5, 000, 000.

21

Table 2.1: Three grids used for the grid dependency study of the RANS (AN-SYS/FLUENT) case for 2D hydrofoil

Grid Original Refined a Refined bCell number 45,878 473,822 1,259,924

To ensure sufficient resolution to capture the boundary layer character-

istics, the value of y+ on the hydrofoil surface should be limited to be less than

5. Fig. 2.9 plots y+ on the hydrofoil using the two refined meshes. Either one

of the two meshes is proven to have sufficient resolution. The refined mesh b

is used for the following study. The information for the RANS case is listed

in Table. 2.2.

(a) Refined grid a (b) Refined grid b

Figure 2.9: y+ along the hydrofoil body subject to a uniform inflow with zeroangle of attack, predicted by RANS (ANSYS/FLUENT) using the refined grida and b, Re = U∞c

υ= 5, 000, 000.

The velocity vectors in the flow domain close to the hydrofoil trailing

edge predicted by RANS are shown in Fig. 2.10. No separation happens in

22

Table 2.2: Information of the RANS (ANSYS/FLUENT) case for 2D hydrofoilReynolds number 5, 000, 000Turbulence model k-ω SST

Pressure correction scheme SIMPLECSpatial discretization Second order upwind

Total time for calculate (8 CPUs) 4,386 seconds

this case. Obvious effects of the boundary layer can be observed in the figure.

To analyze the boundary layer quantitatively, the displacement thickness δ∗ is

introduced, which is defined as

δ∗ =

∫ ∞0

(1− u

Ue

)dy (2.33)

Eight points (shown in Fig. 2.11) are selected on the surface of the hydrofoil.

Along the normal direction of each point, the velocity profile can be exported

from the RANS case, and the displacement thickness at this point can be cal-

culated by (2.33). The velocity profiles at point 2 and point 4 exported from

RANS are plotted in Fig. 2.12. It is reasonable that the thicker boundary layer

is observed at point 2, closer to the trailing edge, than 4. The comparison of

the displacement thickness δ∗ predicted by RANS and the 2D viscous/inviscid

interactive method (shown in Fig. 2.13) shows considerable consistency. Fig.

2.14 plots the pressure distributions along the hydrofoil body predicted by dif-

ferent methods. The pressure predicted by the 2D viscous/inviscid interactive

method shows a better correlation with RANS than the panel method only,

which indicates that the present method can effectively predict the influences

of the boundary layer on pressure. The skin friction coefficient Cf = τw12ρU2

∞is

also compared in Fig. 2.15. The results from RANS (ANSYS/FLUENT) and

23

the present method show similar behavior. Although some differences can still

be noticed in the comparison, the skin friction coefficient Cf predicted by the

present method should be more reasonable Cf than any empirical constant.

Figure 2.10: Velocity vectors close to the hydrofoil trailing edge in a uniforminflow with zero angle of attack, predicted by RANS (ANSYS/FLUENT),Re = U∞c

υ= 5, 000, 000.

24

Figure 2.11: Diagram of the eight selected eight points

Figure 2.12: Velocity profiles at point 2 and 4 on the hydrofoil surface in a uni-form inflow with zero angle of attack, predicted by RANS (ANSYS/FLUENT),Re = U∞c

υ= 5, 000, 000.

25

(a) Pressure side (b) Suction side

Figure 2.13: Comparison of δ∗ on the surface of the hydrofoil subject to auniform inflow with zero angle of attack, between RANS (ANSYS/FLUENT)and the present method, Re = U∞c

υ= 5, 000, 000.

(a) RANS vs. Panel method (b) RANS vs. present method

Figure 2.14: Comparison of Cp on the surface of the hydrofoil subject to auniform inflow with zero angle of attack, between RANS (ANSYS/FLUENT)and the present method, Re = U∞c

υ= 5, 000, 000.

26

Figure 2.15: Comparison of Cf on the surface of the hydrofoil subject to auniform inflow with zero angle of attack, between RANS (ANSYS/FLUENT)and the present method, Re = U∞c

υ= 5, 000, 000.

27

Chapter 3

Three Dimensional Viscous/Inviscid

Interactive Method

In this chapter, the author follows the assumption of Hufford (1992,

1994) that the effects of the boundary layer on the span wise direction is neg-

ligible compared with those on the stream wise direction. For the boundary

layer on propeller blades, it is assumed that the boundary layer radially out-

ward could be ignored, while the boundary layer along the constant radius

needs to be considered. Based on the above assumption, a more reasonable

and complete 3D viscous/inviscid interactive method is developed by coupling

a low-order potential based panel method and the 2D integral boundary layer

analysis. A significant improvement of this method, as compared with the

work of Hufford (1992, 1994) and Sun (2008), is that the viscous effects of

neighboring strips and blades are considered. The present method is intro-

duced in details at first. Then two propellers and a bare duct cases are carried

out. The results predicted by the 3D viscous/inviscid boundary layer method

are presented and correlated with those of RANS cases or experimental mea-

surements.

28

3.1 Assumption

Milewski (1997) developed a 3D fully simultaneous viscous/inviscid in-

teractive scheme, by coupling a potential based panel method and the 3D

integral boundary layer equations. His scheme is totally three dimensional

and involves a large number of unknowns. He solved the whole system using

Newton’s method, which needs to calculate many sensitivities. Thus solving

this scheme requires large amounts of computational resources. This scheme

was only applied to some cases with simple geometries such as hydrofoils or

axis-symmetric ducts. For more complicated geometries such as propeller-

s, this scheme may have more difficulties. For propeller flows particularly, to

avoid solving the 3D integral boundary layer equations, it can be assumed that

the effects of the boundary layer variables radially outward are much smaller

than those along the constant radius. Thus radial effects could be negligible.

In fact, this assumption is reasonable according to the measurements of Jessup

(1989). In addition, the RANS results of Propeller NSRDC4381 (more infor-

mation and results of this propeller will be presented in later sections) in this

thesis also verify the assumption. Fig. 3.1 shows one blade of the propeller.

Define the direction along the constant radius as s and the direction radially

outward as t. The two velocity components in s and t directions along the

normal vector of the blade surface s × t are checked at two strips r/R = 0.6

and r/R = 0.95, so that the effects of the boundary layer in the two directions

could be investigated. Fig. 3.2 and Fig. 3.3 show the velocity profiles at the

middle and end chord (close to trailing edge) of the strip r/R = 0.6 at design

29

J . Close to the tip of the blade r/R = 0.95, the velocity profiles are plotted

in Fig. 3.4 and Fig. 3.5. Fig. 3.6, Fig. 3.7, Fig. 3.8 and Fig. 3.9 plot similar

results at J = 0.5. For the case of J = 0.5, the boundary layer effects along t

direction are very limited and localized compared with those along s direction,

as the assumption states. In the case of design J , the assumption is valid on

the most parts of the blade. However, very close to the blade tip, the effects

of the boundary layer along t are as significant as those of the boundary layer

along s, as shown in Fig. 3.4(b) and Fig. 3.5(b).

r/R=0.9

r/R=0.6

Figure 3.1: Blade geometry of Propeller NSRDC4381

30

height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

Us

Ut𝜹𝒔

𝜹𝒕

(a) Pressure side

height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

Us

Ut𝜹𝒔

𝜹𝒕

(b) Suction side

Figure 3.2: Velocity components in t and s directions at the middle chord of thestrip r/R = 0.6, of propeller 4381, predicted by RANS (ANSYS/FLUENT),Js = 0.889.

height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

Us

Ut𝜹𝒔

𝜹𝒕

(a) Pressure side

height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

Us

Ut

𝜹𝒔

𝜹𝒕

(b) Suction side

Figure 3.3: Velocity components in t and s directions at the end chord (closeto trailing edge) of the strip r/R = 0.6, of propeller 4381, predicted by RANS(ANSYS/FLUENT), Js = 0.889.

31

height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

5

Us

Ut

𝜹𝒔

𝜹𝒕

(a) Pressure side

height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

5

Us

Ut

𝜹𝒔

𝜹𝒕

(b) Suction side


height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

5

Us

Ut

𝜹𝒔

𝜹𝒕

(a) Pressure side

velocity

height

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

5

Us

Ut

𝜹𝒔

𝜹𝒕

(b) Suction side


32

height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

5

Us

Ut

𝜹𝒔

𝜹𝒕

(a) Pressure side

height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

5

Us

Ut

𝜹𝒔

𝜹𝒕

(b) Suction side


height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

5

Us

Ut

𝜹𝒔

𝜹𝒕

(a) Pressure side

height

velocity

0 0.005 0.01 0.015 0.02-1

0

1

2

3

4

5

Us

Ut

𝜹𝒔

𝜹𝒕

(b) Suction side


33

height

height

0 0.005 0.01 0.015-1

0

1

2

3

4

5

6

7

8

Us

Ut

𝜹𝒔

𝜹𝒕

(a) Pressure side

height

velocity

0 0.005 0.01 0.015-1

0

1

2

3

4

5

6

7

8

Us

Ut

𝜹𝒔

𝜹𝒕

(b) Suction side


height

velocity

0 0.005 0.01 0.015-1

0

1

2

3

4

5

6

7

8

Us

Ut

𝜹𝒔

𝜹𝒕

(a) Pressure side

height

velocity

0 0.005 0.01 0.015-1

0

1

2

3

4

5

6

7

8

Us

Ut

𝜹𝒔

𝜹𝒕

(b) Suction side


34

3.2 3D Wall Transpiration Model

Following the assumption mentioned in the above section, a 3D pro-

peller blade is divided into several strips along the constant radius as shown

in Fig. 3.10. The boundary layer along each strip is considered. Similar with

the 2D model, this 3D model also expresses the edge or viscous velocity as the

sum of the inviscid velocity and the induced velocity by blowing sources added

on blades and wakes. Detailed derivation of the expression of the edge velocity

is presented. The equation of the edge velocity should be resolved with the 2D

integral boundary layer equations on the strips. Compared with the 2D case,

the expression of the edge velocity for 3D case contains more unknowns, so

the combined equations cannot be solved directly. Therefore, this 3D model

applies an iterative scheme to solve for the extra unknowns. The procedures

of this scheme is introduced in details.

3.2.1 One Blade

Similar with the 2D case, blowing sources with unknown strengths are

added on the blade and wake panels when the effects of fluid viscosity are con-

sidered. Analogous to (2.24), the viscous or edge velocity on the 3D blade and

the wake has two components: the inviscid velocity and the induced velocity

by the blowing sources. Considering a control point i on a strip m, the viscous

velocity should be written as:

U ei,m =

∂Φi,m

∂s= U inv

i,m +

N+NW∑j=1

Di,j,mmj,m +mr∑

k=1,k 6=m

N+NW∑j=1

Di,j,kmj,k (3.1)

35

where mr is the number of strips. Compared with (2.24), the extra term in

(3.1) represents the induced velocity by the sources on other strips (except

strip m). It should be noticed that the influence coefficient Di,j,k in (3.1) is

different than that in (2.24). In this case, the influence coefficients should be

calculated using the 3D formulations (see appendix), since the panels of each

strip are three dimensional. In (2.24) , however, the influence coefficients are

calculated using the formulations for 2D panels (see appendix).

Figure 3.10: Sketch of 3D blade paneling

The unknown mass defects mj,k in (3.1) should be resolved with the

integral boundary layer equations from (2.29) to (2.32). However, the term∑mrk=1,k 6=m

∑N+NWj=1 Di,j,kmj,k in (3.1) introduces extra unknowns, so that the

combined equations cannot be solved directly. An iterative scheme is intro-

36

duced to solve for all the unknowns. First, the inviscid velocity is calculated

by the 3D low-order potential based panel method. Then the unknown mass

defects in the term∑mr

k=1,k 6=m∑N+NW

j=1 Di,j,kmj,k are initially given as zero to

reduce the number of unknowns. Next, (3.1) can be resolved by the 2D integral

boundary layer solver used in XFOIL on each strip independently. Based on

the solved mass defects, the term∑mr

k=1,k 6=m∑N+NW

j=1 Di,j,kmj,k is updated and

added into the inviscid velocity term U invi,m . Solve for the mass defects on each

strip again. Repeat the above procedures until a converged result is obtained.

The brief solution procedures of this scheme are shown in Fig. 3.11.

3.2.2 Multiple Blades

When multiple blades are considered, (3.1) becomes:

U ei,m,1 =

∂Φi,m,1

∂s

= U invi,m,1 +

N+NW∑j=1

Di,j,m,1mj,m,1 +mr∑

k=1,k 6=m

N+NW∑j=1

Di,j,k,1mj,k,1

+NB∑kk=2

mr∑k=1

N+NW∑j=1

Di,j,k,kkmj,k,kk

(3.2)

where NB is the number of blades. The blade kk = 1 is the key blade. The

term∑NB

kk=2

∑mrk=1

∑N+NWj=1 Di,j,k,kkmj,k,kk represents the induced velocity by

the sources added on other blades (except the key blade). For steady cases,

the boundary layer variables on other blades are the same with those on the

key blade, for instance, mj,m,kk = mj,m,1. Thus (3.2) can be rewritten as the

37

The inviscid velocity 𝑈𝑖𝑛𝑣

(low-order potential based panel method)

Update the induced velocity

from other strips

𝐷𝑖,𝑗,𝑘𝑚𝑗,𝑘𝑁+𝑁𝑤𝑗=1

𝑚𝑟𝑘=1,𝑘≠𝑚

Update the inviscid velocity

𝑈𝑖𝑛𝑣

Initially 𝐷𝑖,𝑗,𝑘 = 0

Solve for mass defects 𝑚𝑗,𝑘

on each strip

Converged?

No

Yes Final results

Figure 3.11: Flowchart of the iterative scheme for 3D viscous/inviscid bound-ary layer method

38

following:

U ei,m,1 =U inv

i,m,1 +

N+NW∑j=1

Di,j,m,1mj,m,1 +NB∑kk=2

N+NW∑j=1

Di,j,m,kkmj,m,kk

+mr∑

k=1,k 6=m

N+NW∑j=1

(Di,j,k,1 +

NB∑kk=2

Di,j,k,kk

)mj,k,1

(3.3)

(3.3) is very useful for the simplification of coding. Using the similar proce-

dures shown in Fig. 3.11, all the unknowns in (3.3) can be resolved. In fact,

the effects from other blades are very limited. Fig. 3.12 compares the pressure

along two different strips, r/R = 0.6 and r/R = 0.9, on the key blade with

and without the effects of other blades of Propeller NSTDC 4381 at design

condition Js = 0.889. The comparisons prove the above statement about the

limited effects of other blades.

3.3 Propeller DTMB4119

Propeller DTMB4119 is a 3 bladed propeller (shown in Fig. 3.13). The

design advance ratio of this propeller Js is 0.833. Jessup (1989) investigated

the pressure and the boundary layer characteristics on the propeller blades

experimentally. The 3D viscous/inviscid interactive method described above

is applied to this propeller, and the results are correlated with the experimental

data from Jessup (1989). Before any further investigation, a grid dependency

study is conducted to validate the present method. Both the numbers of panels

along the chord wise and span wise directions are changed. Fig. 3.14 plots

the pressure distributions along the strip at r/R = 0.62. Despite the changes

39

(a) r/R=0.6 (b) r/R=0.9

Figure 3.12: Pressure distributions along two different strips, r/R=0.6 andr/R=0.9, of Propeller NSTDC 4381, predicted by the present method with andwithout the effects of blowing sources added on neighboring blades, Js = 0.889,Re = U∞D

υ= 7.42e5, 1% turbulence level, fixed transition points at 0.1 chord

length on both the pressure and suction sides of each strip.

40

of the panel numbers, the plots of the pressure distribution predicted by the

present method collapse into the same line, indicating that the method and

the code work fine.

The iterative scheme (shown in Fig. 3.11) is applied to solve for all the

unknowns. Fig. 3.15 shows the pressure distributions at different iterations

along the strip at r/R = 0.62. The plots of the pressure at the 4th and 5th

iteration collapse with each other, indicating that a converged result is already

obtained at the 5th iteration. It should be mentioned that the result of the

first iteration is the one (Hufford’s result) that does not consider the effects of

blowing sources on neighboring strips and blades.

Fig. 3.16 and Fig. 3.17 compare the displacement thickness δ∗ along

Figure 3.13: Paneled geometry of Propeller DTMB4119

41

Figure 3.14: Pressure distributions predicted by the present method using dif-ferent numbers of panels along the strip r/R = 0.62 of Propeller DTMB4119,Js = 0.833, Re = U∞D

υ= 766, 395, 1% turbulence level, fixed transition points

at 0.1 chord length on both the pressure and suction sides of each strip.

42

(a) Full view (b) Details close to the trailing edge

Figure 3.15: Pressure distributions at different iterations on the strip r/R =0.62 of Propeller DTMB4119, predicted by the present method, Js = 0.833,Re = U∞D

υ= 766, 395, 1% turbulence level, fixed transition points at 0.1 chord

length on both the pressure and suction sides of each strip.

the strip at r/R = 0.7 predicted by the present method with the experimental

measurements. The comparison on the suction side shows considerable consis-

tency. On the pressure side, even though the experimental data is insufficient

for detailed comparison, the results predicted by the present method and the

experimental measurements show a same trend. The comparisons of the pres-

sures along different strips between the experiment and the present method

are shown from Fig. 3.18 to Fig. 3.20. Considering the difficulties and the

errors of the experimental measurements, the comparisons shown in the figures

are acceptable.

For 3D propeller blade cases, the prediction of the thrust/torque forces

at various operating conditions is a main concern. The forces consist of two

43

Figure 3.16: Comparison of displacement thickness on the pressure side of thestrip r/R = 0.7 of Propeller DTMB4119 between experiments and the presentmethod (free transition point on the pressure side of each strip, 1% turbulencelevel), Js = 0.833, Re = U∞D

υ= 766, 395.

Figure 3.17: Comparison of displacement thickness on the suction side of thestrip r/R = 0.7 of Propeller DTMB4119 between experiments and the presentmethod (fixed transition point at 0.5 chord length on the suction side of eachstrip, 1% turbulence level), Js = 0.833, Re = U∞D

υ= 766, 395.

44

Figure 3.18: Comparison of pressure coefficients on the strip, r/R = 0.3 of,of Propeller DTMB4119 between experiments and the present method (fixedtransition point at 0.5 chord length on the suction side and free transitionpoint on the pressure side of each strip, 1% turbulence level), Js = 0.833,Re = U∞D

υ= 766, 395.

45


υ= 766, 395.

46


υ= 766, 395.

47

parts: one part is due to the friction on propeller blades and the other part is

due to the pressure. Panel methods ignore the effects of fluid viscosity, thus

cannot take the friction into consideration and cannot predict the influences

of boundary layer on pressure. An easy solution of this dilemma is to apply

empirical methods to consider the effects of fluid viscosity. However, such

method requires user adjustable corrections that are not general for all cases.

It has been proved that the viscous/inviscid interactive method can precisely

predict the changes of pressure on the wall due to boundary layer. For the

friction part, instead of using the empirical constants, the 3D viscous/inviscid

interactive method can predict more reasonable skin friction coefficient Cf

on propeller blades. Then the forces due to the friction are calculated by

integrating the shear stress τw = Cf · 12ρU2∞ over the blade surfaces. Fig. 3.21

shows the comparisons of KT and KQ between the experimental measurements

and the present method. The results predicted by the present method exhibit

considerable agreement with the experimental data.

3.4 Propeller NSRDC4381

The 3D viscous/inviscid interactive method is also applied to Propeller

NSRDC4381, which is a five-bladed propeller (shown in Fig. 3.22). The design

advance ratio Js of this propeller is 0.889. The results predicted by the present

method and RANS (ANSYS/FLUENT) simulations conducted by Tian and

Kinnas (2011) are compared. Fig. 3.23 shows the domain and boundary con-

ditions for RANS (ANSYS/FLUENT) simulations. The grid details are shown

48

Figure 3.21: Comparisons of KT and KQ of Propeller DTMB4119 at differentadvance ratios, between experiments and the present method.

49

in Fig. 3.24. Some important information of the RANS (ANSYS/FLUENT)

cases is listed in Table. 3.1.

Figure 3.22: Paneled geometry of Propeller NSRDC4381.

For the present 3D viscous/inviscid interactive method, it is necessary

to conduct a grid dependency study at first. The pressure distributions with

different numbers of panels along two strips on the propeller blade are com-

pared in Fig. 3.25. The great consistency shown in Fig. 3.25 indicates that the

changes of the number of panels on both chord and span wise directions do not

affect the final results. The pressure distributions at different iterations are

checked as well (shown in Fig. 3.26) to ensure the convergence of the results.

The 3D viscous/inviscid interactive method is applied to predict the

pressure on the blades of propeller 4381. Fig. 3.27 to Fig. 3.30 shows detailed

50

comparisons of pressure at two different operating conditions: the design con-

dition (Js = 0.889) and an off-design condition (Js = 0.5). The pressures

predicted by the present method show better correlation with the RANS re-

sults than those predicted by the panel method only, especially at the region

close to the blade trailing edge. Closer to the trailing edge, the effects of flu-

id viscosity become more significant and the boundary layer becomes thicker.

Thus more clear improvements of the results are observed in this region with

the boundary layer correction.

The thrust/torque forces due to the fluid viscosity is calculated by in-

tegrating the shear stress τ over the blade surfaces. Fig. 3.31 shows the

comparison of Cf predicted by the present method and RANS at the design

Figure 3.23: Domain and boundary conditions of the RANS (AN-SYS/FLUENT) case for Propeller NSRDC4381 (from Sharma 2011).

51

Figure 3.24: Grid details used in the RANS (ANSYS/FLUENT) case for Pro-peller NSRDC4381. Top left: O type grid on the propeller blade. Top right:grid details on the hub around the root section of the propeller blade. Bot-tom left: Grid details on the hub near the leading edge of the root section ofpropeller blade. Bottom right: grid details on the hub near the trailing edgeof the root section of the propeller blade. (from Sharma 2011).

52

(a) r/R=0.6 (b) r/R=0.9

Figure 3.25: Pressure distributions predicted by the present method usingdifferent numbers of panels at strips, r/R = 0.6 and r/R = 0.9, of PropellerNSRDC4381, Js = 0.889, Re = U∞D

υ= 7.42e5, 1% turbulence level, fixed

transition points at 0.1 chord length on both the pressure and suction sides ofeach strip.

53

Table 3.1: Information of the RANS (ANSYS/FLUENT) case for PropellerNSRDC4381

Cell number 1, 476, 540Reynolds number (J=0.889) 6.22e5Reynolds number (J=0.5) 4.15e5

Turbulence model k-ω SSTPressure correction scheme SIMPLEC

Spatial discretization QUICKTotal time for calculate (32 CPUs) 4,386 seconds

condition Js = 0.889. The comparison exhibits an acceptable agreement, in-

dicating that the present method could predict the friction on the propeller

blades. The plots of KT and KQ of this propeller at different operating condi-

tions are shown in Fig. 3.32. Compared with the experimental measurements,

(a) r/R=0.6 (b) r/R=0.9

Figure 3.26: Pressure distributions at different iterations at strips r/R = 0.6and r/R = 0.9 of Propeller NSRDC4381, predicted by the present method,Js = 0.889, Re = U∞D

υ= 7.42e5, 1% turbulence level, fixed transition points

at 0.1 chord length on both the pressure and suction sides of each strip.

54

(a) Full view (b) Detailed view close to trailing edge

Figure 3.27: Comparison of pressure distributions along the strip, r/R =0.6, of Propeller NSRDC4381, predicted by RANS (ANSYS/FLUENT), panelmethod and the present method (fixed transition points at 0.1 chord lengthon both the pressure and suction sides of each strip, 1% turbulence level),Js = 0.889, Re = U∞D

υ= 7.42e5 for viscous cases.

55




56




57




58

the results predicted by the present method behave well near the design con-

dition, but under-estimate the KT and KQ at low Js (blow Js = 0.5). In fact,

at low J conditions, the effects of the wake become more significant. Thus

more accurate wake alignment schemes are needed. The results shown in Fig.

3.32 are obtained by the present method combined with PSF2 wake alignmen-

t. Kinnas et al. (2012) developed a new full wake alignment scheme FWA.

The results predicted by the present method with the full wake alignment

scheme are shown in Fig. 3.33, from which obvious improvements at low J are

observed.

(a) r/R=0.6 (b) r/R=0.74

Figure 3.31: Comparison of Cf at strips, r/R = 0.6 and r/R = 0.6, of PropellerNSRDC4381, between RANS (ANSYS/FLUENT) and the present method(fixed transition points at 0.1 chord length on both the pressure and suctionsides of each strip, 1% turbulence level), Js = 0.889, Re = U∞D

υ= 7.42e5.

59

Figure 3.32: Comparisons of KT and KQ of Propeller NSRDC4381 at differ-ent advance ratios, between the present method combined with PSF2 wakealignment and experiments.

60

Figure 3.33: Comparisons of KT and KQ of Propeller NSRDC4381 at differentadvance ratios, between the present method combined with full wake alignmentand experiments.

61

3.5 DTMB Duct II

The 3D viscous/inviscid interactive method is applied to a bare duct

case. DTMB Duct II is an axis-symmetric duct, which has a NACA66 thick-

ness form, with a maximum thickness-chord ratio tmax/C = 0.1. The camber

has a NACA a = 0.8 mean distribution and the maximum camber-chord ra-

tio is fmax/C = 0.04. Fig. 3.34 shows the paneled geometry of this duct.

The duct is placed at zero angle of attack. The Reynolds number Re = U∞Dν

is 2.06 × 106. The pressure on the duct predicted by the present method is

compared with the measurements by Morgan and Caster (1965).

Figure 3.34: Paneled geometry of the DTMB Duct II.

The result (pressure) of a grid dependency study is shown in Fig. 3.35.

Using different numbers of panels on the chord and span wise directions, the

62

plots of the pressures predicted by the current method collapse into the same

line. Fig. 3.36 shows the convergence process of the pressure. At the 5th

iteration, converged results are obtained. Since DTMB Duct II is an axis-

symmetric duct and the angle of attack is zero, it is expected that the pressure

or the boundary layer variables be the same at different sections along the

chord wise direction. The pressure and the displacement thickness of boundary

layer at three selected strips along chord wise direction are checked and shown

in Fig. 3.37 and Fig. 3.38, respectively.

x/c

-Cpvis

0 0.2 0.4 0.6 0.8

-0.8

-0.6

-0.4

-0.2

0

0.2

60 X 90

60 X 100

80 X 100

Figure 3.35: Pressure distributions on DTMB Duct II predicted by the presentmethod using different numbers of panels on both the chord and span wisedirections, Re = U∞D


both the pressure and suction sides of each strip, 1% turbulence level.

Fig. 3.39 shows the predicted invsicid (panel method only) and viscous

63

Figure 3.36: Pressure distributions at different iterations on DTMB Duct II,predicted by the present method, Re = U∞D

υ= 2.06e6, fixed transition points

at 0.05 chord length on both the pressure and suction sides of each strip, 1%turbulence level.

64

Figure 3.37: Pressure distributions along three different sections of DTMBDuct II, predicted by the present method, Re = U∞D


points at 0.05 chord length on both the pressure and suction sides of each strip,1% turbulence level.

65

Figure 3.38: Displacement thickness of boundary layer along three differentsections of DTMB Duct II, predicted by the present method, Re = U∞D

υ=

2.06e6, fixed transition points at 0.05 chord length on both the pressure andsuction sides of each strip, 1% turbulence level.

66

(present method) pressure on the duct strip compared with the experimental

measurements from Morgan and Caster (1965). Both the viscous and inviscid

solution show good correlations with the experiment data.

Figure 3.39: Viscous and inviscid pressure distributions on the DTMB Duct IIpredicted by the present method (fixed transition points at 0.05 chord lengthon both the pressure and suction sides of each strip, 1% turbulence level),compared with the experimental measurements by Morgan and Caster (1965),Re = U∞D

υ= 2.06e6.

67

Chapter 4

Viscous Image Model

In this chapter, the image model used to simulate the effects of slip

walls developed by Singh (2009) is introduced. The author further improves

this model for viscous cases. A case of a straight wing blade between two

slip walls is carried out, and the results are compared with those of a RANS

case. Next, the results of a case of a swept wing blade between two slip ways

are presented and analyzed. The slip walls exclude the effects of the blade

tip, so that the 3D viscous/inviscid interactive is more easy to be applied.

In addtion, the case of the straight wing between two parallel slip walls is

expected equivalent with a 2D case, which is a good test to validate the 3D

viscous/inviscid interactive method.

4.1 Image Model with Panel Method Only

Singh (2009) introduced an image model to simulate the effects of slip

walls. For instance, the system of a wing blade adjacent to a slip wall shown

in Fig. 4.1 is equivalent to the one shown in Fig. 4.2. In Fig. 4.2, the effects

of the slip wall are considered by mirroring the wing blade about the wall, so

that the derivative of any variable cross the red line is zero as the effects of

68

the slip wall in Fig. 4.1.

Figure 4.1: Diagram of a straight wing blade adjacent to a slip wall

Figure 4.2: Diagram of a straight wing blade and its images, symmetric aboutthe red line

Since the wing blade and its image shown in Fig. 4.2 are symmetric

about the wall, the potential on a panel of the wing blade should be the same

69

with that on the corresponding image panel symmetric about the wall. With

this statement, the inviscid solution of this case can be obtained by solving

(2.12), but with new influence coefficients to consider the effects of the image.

The new influence coefficients are expressed as:

Anewij = Aij + Aij′ (4.1)

Snewij = Sij + Sij′ (4.2)

where j′ is the image panel corresponding to the original panel j on the wing

blade. One useful fact is that the source or dipole influence coefficient due to

an image panel at a particular control point is the same as the one due to the

original panel at the image control point (shown in Fig. 4.3). According to

this fact, the influence coefficients can be rewritten as:

Anewij = Aij + Ai′j (4.3)

Snewij = Sij + Si′j (4.4)

The new expressions are very useful for the simplification of coding. When

multiple images are considered, the influence coefficients become:

Anewij = Aij +

Image∑i′

Ai′j (4.5)

Snewij = Sij +

Image∑i′

Si′j (4.6)

where Image is the number of images. With the new influence coefficients,

the inviscid solution can be obtained by solving (2.12).

70

Figure 4.3: Diagram of the equivalence of the influence coefficients, from Singh(2009)

4.2 Image Model with 3D Viscous/Inviscid InteractiveMethod

Once the inviscid solution is obtained, blowing sources with unknown

strengths are added on the panels of the original wing blade, the wake and

their images to consider the effects of fluid viscosity. Analogous to (3.1), the

viscous velocity becomes:

U ei,m =

∂Φi,m

∂s

= U invi,m +

N+NW∑j=1

Di,j,mmj,m +mr∑

k=1,k 6=m

N+NW∑j=1

Di,j,kmj,k

+

Imag∑im=1

mr∑k=1

N+NW∑j=1

Di,j,k,immj,k,im

(4.7)

71

where Image is the number of images. Using the same iterative scheme de-

scribed in Fig. 3.11, The unknown mass defects in (4.7) can be resolved. In

this case, the last two terms on the right hand side of (4.7) should be updated

each iteration and added to the inviscid velocity term.

4.3 Straight Wing between Two Parallel Slip Walls

A case of a 3D straight wing blade between two parallel walls (shown in

Fig. 4.4) subject to a uniform inflow with a 2 degree angle of attack is carried

out. For this case, infinite images are required on both sides to simulate the

effects of the two parallel walls. Practically, a reasonable number of images

can be determined through convergence studies to obtained fairly accurate

numerical results.

Figure 4.4: Diagram of a wing blade between two parallel walls

The result (pressure) of a grid dependence study is presented in Fig.

4.5. The plots of the pressure coefficients along the strip at r/R = 0.75

obtained using different numbers of panels on both span wise and chord wise

72

directions collapse into the same line. In other words, the changes of the

number of panels do not affect the final result, indicating a good convergence

character of the present method and the code.

Figure 4.5: Pressure distributions along the strip at r/R = 0.75, predicted bythe present method using different numbers of panels on span and chord wisedirections, Re = U∞c


both the pressure and suction sides of each strip, 1% turbulence level.

A convergence study is conducted to determine a reasonable number of

images to simulate the effects of the two parallel slip walls. Different numbers

of images on each side are selected, and the predicted circulations are plotted

in Fig. 4.6, from which the circulation is converging with the increase of the

image number. The viscous and invscid pressure distributions along the strip

73

at r/R = 0.6 on the wing blade are plotted in Fig. 4.7. For the convergence

of pressure, 10 images each side are sufficient.

Figure 4.6: Circulations on the straight wing blade predicted by the originalimage model of Singh (2009) using different numbers of images.

A case of a 3D straight wing between two parallel walls subject to a

uniform inflow with a particular angle of attack is expected equivalent to a 2D

wing strip subject to the same inflow. In other words, the flow around the 3D

wing blade could be reduced to a 2D flow around a 2D strip extracted from

the wing blade along the chord wise direction. Thus it is expected that the

pressure distributions at different strips of the straight wing blade should be

the same. Using 10 images each side, the pressure distributions along three

different strips are plotted in Fig. 4.8 for both viscous (a) and invscid (b)

74

(a) Viscous (b) Inviscid

Figure 4.7: Pressure distributions along the strip at r/R = 0.6 of the straightwing blase, predicted by the original and the present image model, using d-ifferent numbers of images. For the viscous case, Re = U∞c

υ= 1.0e6, fixed

transition points at 0.1 chord length on both the pressure and suction sides ofeach strip, 1% turbulence level.

75

cases. The plots of the pressure distributions collapse into the same line, as

expected.

(a) Viscous (b) Inviscid

Figure 4.8: Pressure distributions on the strips at r/R = 0.3, r/R = 0.5and r/R = 0.9 of the straight wing blade, predicted by the original and thepresent image model, using different numbers of images. For the viscous case,Re = U∞c

υ= 1.0e6, fixed transition points at 0.1 chord length on both the

pressure and suction sides of each strip, 1% turbulence level.

A 2D and a 3D RANS case are carried out as references. The domain for

the 3D RANS case is shown in Fig. 4.9. H type grid topology is used around

the wing blade as shown in Fig. 4.10. The symmetry boundary condition

is applied to the side boundaries ABC and DEF to simulate the parallel slip

walls. The outlet ABFD is set as the outflow. The rest boundaries are set

as the velocity inflow with the a specified velocity. The value of y+ on the

wing surface is ranged between 30 and 50. Some related information about

the 3D RANS case is listed in Table 4.1. A section along the chord wise

76

direction is extracted from the wing blade. The 2D RANS case for the section

uses the similar mesh and schemes of the 3D case. The pressures at different

strips of the 3D RANS case are compared with the pressure on the 2D section

(shown in Fig. 4.11). The comparison shows great consistency, which proves

the previous statement about the equivalence of the 2D and 3D cases. Fig.

4.12 compares the pressure distributions on the wing blade, predicted by the

RANS, the present method and the panel method only. Both the present

method and the panel method can predict acceptable results, but the pressure

predicted by the present method has better correlation with the RANS result,

especially at the region close to the blade trailing edge.

A

B

C

D

E

F

Figure 4.9: Diagram of the domain of the straight wing blade between twoparallel slip walls for 3D RANS (ANSYS/FLUENT) simulations.

77

Y X

Z

Figure 4.10: Grid details around the straight wing blade used in 3D RANS(ANSYS/FLUENT) simulations.

78

Figure 4.11: Pressure coefficients on the strips at r/R=0.2 and r/R=0.6 ofthe 3D straight wing predicted by the 3D RANS (ANSYS/FLUENT) case,and on the 2D strip predicted by the 2D RANS (ANSYS/FLUENT) case,Re = U∞c

υ= 1.0e6.

79

Figure 4.12: Pressure coefficients along the strip at r/R=0.5 of the 3D straightwing predicted by RANS (ANSYS/FLUENT), the present method (fixed tran-sition points at 0.1 chord length on both the pressure and suction sides ofeach strip, 1% turbulence level) and the panel method. For the viscous case,Re = U∞c

υ= 1.0e6.

80

Table 4.1: Information of the RANS (ANSYS/FLUENT) case for a straightwing blade between two parallel slip walls.

Cell number 537, 080Reynolds number 1e6Turbulence model k-ω SST

Pressure correction scheme SIMPLECSpatial discretization Second order upwind

Total time for calculate (32 CPUs) 20 mins

4.4 Swept Wing between Two Parallel Slip Walls

A case of a 3D swept wing blade between two parallel walls shown

in Fig. 4.13 subject to a uniform flow with a 2 degree angle of attack is

carried out. The pressures along different strips are plotted in Fig. 4.14. The

differences among the plots imply that the case is not equivalent to a simple 2D

strip case. The pressures predicted by the present method are also correlated

with RANS results in Fig. 4.15, Fig. 4.16 and Fig. 4.17. The comparisons

show great agreements.

81

Figure 4.13: Diagram of a swept wing between two parallel slip walls.

Xd/C

-Cp

0.2 0.4 0.6 0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1r/R=0.3

r/R=0.6

r/R=0.9

(a) Viscous

Xd/C

-Cp

0.2 0.4 0.6 0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 r/R=0.3

r/R=0.6

r/R=0.9

(b) Inviscid

Figure 4.14: Pressure distributions on the strips at r/R = 0.3, r/R = 0.5 andr/R = 0.9 of the wing blade, predicted by the present method, Re = U∞c

υ=

1.0e6, fixed transition points at 0.1 chord length on both the pressure andsuction sides of each strip, 1% turbulence level.

82

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

RANS

Viscous Image Model

Inviscid Image Model

r/R=0.025

Figure 4.15: Pressure coefficients on the strip at r/R=0.025 of the 3D swep-t wing, predicted by the 3D RANS (ANSYS/FLUENT) case, the presentmethod (fixed transition points at 0.1 chord length on both the pressure andsuction sides of each strip, 1% turbulence level) and the panel method. Forthe viscous case, Re = U∞c

υ= 1.0e6.

83

x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

RANS

Viscous Image model


r/R=0.5


υ= 1.0e6.

84

x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

1

1.5

2

RANS

Viscous Image Model


r/R=0.975


υ= 1.0e6.

85

Chapter 5

Hydrofoils and Propellers with Non-Zero

Trailing Edge Thickness

In this chapter, the extension scheme for 2D hydrofoils with non-zero

thickness trailing edges or open trailing edges proposed by Pan (2009, 2011)

is further improved. First, a new extension and search scheme in 2D are

introduced in details. Then results of two sample cases are presented and

correlated with those of RANS cases. A similar method for 3D blades with

non-zero trailing edge thickness is introduced next. The method is applied to

a 3D propeller as a sample case and some preliminary inviscid (panel method

only) results are presented.

5.1 2D Hydrofoils with Non-Zero Trailing Edge Thick-ness

5.1.1 Open Trailing Edge Extension Scheme

Panel methods have difficulties dealing with hydrofoils or propeller

blades with non-zero thickness trailing edges. In one instance, Fig. 5.1 shows

the trailing edge of a strip extracted from the ONR-AxWj-2 rotor blade. Gen-

erally, such a trailing edge with finite thickness needs to be closed (shown in

Fig. 5.2) for panel methods. However, for the trailing edge with large thick-

86

ness, this method may change the original geometry too much. Pan (2009,

2011) proposed a scheme that extends the trailing edge with non-zero thick-

ness to form a sharp one (shown in Fig. 5.3).

Finite thickness

Figure 5.1: Section with a non-zero thickness trailing edge, extracted from theONR-AxWj-2 rotor blade

Given the last camber point, the method by Pan (2009, 2011) gener-

ates a new camber and thickness distributions by interpolation of the original

foil and the last camber point. One flaw of this scheme is that the original

geometry may be changed once an inappropriate last camber point is given.

A new extension scheme that uses second order polynomials to describe the

extension edges (as shown in Fig. 5.4) is introduced to overcome the problem

of Pan (2009, 2011). Define the two end points of the upper and lower side

of the open trailing edge as the cut points as shown in Fig. 5.4. The new

scheme estimates the last camber point based on the slopes of the two cut

points, so that inappropriate input point can be avoided. Then two second

87

Figure 5.2: Diagram of the modification of the trailing edge with non-zerothickness

Figure 5.3: Diagram of the extension after the open trailing edge, from Pan(2009)

88

order polynomials are determined to describe the upper and lower edges of the

extension. Suppose the lower side BC of the extension (shown in Fig. 5.4)

can be described as:

y = ax2 + bx+ c (5.1)

where a, b and c are constant unknowns. The three constants should satisfy

the following conditions:

yB = ax2B + bxB + c

yC = ax2C + bxC + c(dy

dx

)B

= 2axB + b

(5.2)

where the third equation ensures the continuity at point B. The expression of

the lower side BC of the extension can be determined by solving (5.2). The

upper side can be determined similarly.

5.1.2 Search Scheme for Extension

Since the extension after the open trailing edge is an artificial part, it

should satisfy the two following conditions:

• No-lift condition

The lift of the extension part should be zero.

• Pressure equivalence condition

The pressures of the cut points should be same.

In fact, the second condition is the primary one to be satisfied. The first con-

dition is always satisfied with the second one. To satisfy the second condition,

89

the estimated last camber point is moved vertically to search the correct po-

sition. Assume that the pressure difference ∆p between the two cut points is

a function of the vertical position of the last camber point yC .

∆p = f(yC)

d(∆p) =df(yC)

dyCdyC

(5.3)

where the derivative df(yC)dyC

can be calculated numerically. The search scheme

is shown in Fig. 5.5. First, the last camber point based on the slopes of the

cut points is estimated. Then extend the trailing edge with the new scheme

to form a sharp one and calculate the pressure difference ∆p between the cut

points. Next, update the vertical position of the last camber point yC by (5.3)

and extend the trailing edge again. Repeat this process until ∆p becomes zero.

Cut points

C

B

Figure 5.4: Diagram of extension of the trailing edge with non-zero thickness

90

Estimate the last camber point

Calculate Δ𝑝 of the cut

points

Update the vertical position

of the last camber point 𝑦𝐶

Initially 𝑈𝑒 = 𝑈𝑖𝑛𝑣

CalculateΔ𝑝 of the cut points

Δ𝑝=0?

No

Yes Final results

Figure 5.5: Diagram of the procedure of the search scheme for extension

5.2 Sample Cases of 2D Hydrofoils

5.2.1 Strip of ONR-AxWj-2 Rotor Blade

The method described above is applied to a 2D strip extracted from

the ONR-AxWj-2 rotor blade at r/R = 0.7. The pressure predicted by the

91

present method has a great correlation with RANS results, shown in Fig. 5.6.

The final pressure difference of the two cut points ∆p is 1.95e − 3, and the

lift on the extension part is less than 0.3 percent of the total lift on the strip,

which satisfies both the zero lift and the pressure equivalence conditions. The

final extension of the trailing edge is shown in Fig. 5.7. The recorded pressure

difference ∆p of the two cut points and the vertical position of the last camber

point yC shows a linear relationship in Fig. 5.8.

Figure 5.6: Comparison of pressure distributions on the strip extractedfrom the ONR-AxWJ-2 rotor blade at r/R = 0.7, between RANS (AN-SYS/FLUENT) and the present method (fixed transition points at 0.01chord length on both the pressure and suction sides, 1% turbulence level),Re = U∞c

υ= 5.0e6.

92

Figure 5.7: Final extension behind the the strip extracted from the ONR-AxWJ-2 rotor blade at r/R = 0.7, predicted by the present method, Re =U∞cυ

= 5.0e6, fixed transition points at 0.01 chord length on both the pressureand suction sides, 1% turbulence level.

93

Figure 5.8: Pressure difference ∆p of the cut points vs. vertical position ofthe last camber point yC , of the ONR-AxWJ-2 rotor blade at r/R = 0.7, fromthe present method, Re = U∞c


length on both the pressure and suction sides, 1% turbulence level.

94

5.2.2 NACA Hydrofoil

Another sample case is a 2D NACA hydrofoil ( fmax/c = 0.02 , tmax/c =

0.04) with a vertical cut at the 90 percent chord length treated as a trailing edge

with non-zero thickness. The pressure difference ∆p of the final result is 3.19e−

4. The lift of the extension part is around 0.5 percent of the total lift. Fig. 5.9

compares the pressure along the section predicted by the present method and

RANS. The great agreement of the comparison validates the present method.

The final extension behind the open trailing edge is shown in Fig. 5.10, and

the linear relationship between ∆p and yC is shown in Fig. 5.11.

Figure 5.9: Comparison of pressure distributions along the NACA hydrofoilwith a non-zero thickness trailing edge, between RANS (ANSYS/FLUENT)and the present method (fixed transition points at 0.01 chord length on boththe pressure and suction sides, 1% turbulence level), Re = U∞c

υ= 9.0e6.

95

Figure 5.10: Final extension behind the NACA hydrofoil with a non-zerothickness trailing edge, predicted by the present method, Re = U∞c

υ= 5.0e6,

fixed transition points at 0.01 chord length on both the pressure and suctionsides, 1% turbulence level.

96

Figure 5.11: Pressure difference ∆p of the cut points vs. vertical position of thelast camber point yC , of the NACA hydrofoil with a non-zero thickness trailingedge, from the present method, Re = U∞c

υ= 5.0e6, fixed transition points at

0.01 chord length on both the pressure and suction sides, 1% turbulence level.

97

5.3 Propeller Blades with Non-zero Trailing Edge Thick-ness

For 3D propeller blades with non-zero thickness trailing edges, each

blade is divided into several strips along the constant radius. The trailing

edge of each strip is extended using the extension scheme used in 2D hydrofoil

cases. Next, the Newton’s method is used to search the correct positions of

the last camber points of the extensions so that the no-lift and the pressure

equivalence (primary one) conditions are satisfied. In 3D cases, the pressure

difference ∆p between the cut points of one section is a function of the vertical

positions of the last camber points of all the sections (yc,1, yc,2 . . . yc,mr).

∆p1 = f1 (yc,1, yc,2 . . . yc,mr)

∆p2 = f2 (yc,1, yc,2 . . . yc,mr)

. . .

∆pmr = fmr (yc,1, yc,2 . . . yc,mr)

(5.4)

where mr is the number of strips on a blade. (5.4) leads to the following

relationship:

d

∆p1∆p1. . .

∆pmr

=

∂f1∂yc,1

∂f1∂yc,2

. . . ∂f1∂yc,mr

∂f2∂yc,1

∂f2∂yc,2

. . . ∂f2∂yc,mr

. . . . . . . . . . . .∂fmr∂yc,1

∂fmr∂yc,2

. . . ∂fmr∂yc,mr

d

yc,1yc,2. . .yc,mr

(5.5)

The Jacobin matrix in (5.5) could be calculated numerically. The last camber

points of the strips are moved vertically a small amount one by one. Each time,

the pressure difference between the cut points of each strip is evaluated. The

derivatives in the matrix are calculated by dividing the corresponding pressure

98

differences by the small movement. Once the initial positions of the last camber

points of the strips are estimated, the pressure difference ∆p between the cut

points of each strip could be evaluated. Then the positions of the last camber

points of all the strips are updated following (5.5). Repeat the process until

the pressure equivalence condition is satisfied on all the strips.

A five-bladed propeller with non-zero thickness trailing edge (studied

in Pan 2009) is used in this study. The details of the propeller geometry are

restricted to show in this thesis. A global view of this propeller is shown in

Fig. 5.12. The extension scheme described in the 2D case is applied to each

strip of the propeller blade. The initial modified geometry is shown in Fig.

5.13.

Figure 5.12: Propeller with non-zero trailing edge thickness, from Pan (2009).

99

The last camber points of the strips are moved a small amount vertically

one by one to calculate changes of the pressure difference between the cut

points of each strip. Then the derivatives in the Jacobin matrix in (5.5) are

calculated numerically. Fig. 5.14 shows the changes of the pressure difference

between the cut points on strip1, strip 2 and strip 3 due to the movement of

the last camber point on strip 1, at Js = 0.75. Fig. 5.15, Fig. 5.16 and Fig.

5.17 show similar results. All these results indicate that a small movement of

the last camber point of one strip could cause a linear change of ∆p between

the cut points of the strip itself and neighboring strips.

Following (5.5), the positions of the last camber points of all the strips

are updated until the pressure equivalence condition is satisfied on all the

closed sharp trailing edge

Figure 5.13: Propeller with closed sharp trailing edge modified by the presentscheme.

100

Figure 5.14: Pressure differences between the cut points of strip 1 and neigh-boring strips vs. the movement of the last camber point of strip 1.


101



102

strips. The pressure difference between the cut points of each strips is plotted

in Fig. 5.18. The blue line represents ∆p on the initial extension. The red

line represents the results on the final extension (after several updates of the

last camber points). The pressure equivalence condition is satisfied on most

strips after four iterations. For the strips very close to the hub and the tip,

however, the pressure differences between the cut points do not decrease very

obviously, this is because the movements of the last camber points on these

strips are restrained to avoid abrupt changed along the trailing edge curve. The

circulation on the final blade geometry is plotted in Fig. 5.19. The smooth

curve of the circulation indicates the final blade geometry is acceptable and

ready for further studies. However, it should be noticed that the success of the

3D search scheme in this case will not necessarily guarantee the success in other

cases. Because the derivatives in the Jacobin matrix calculated numerically

are only accurate when the movements of the last camber points are restrained

in a small amount around the initial positions. In other cases (different Js),

the 3D search scheme may lead some large movements of the last camber

points. The changes of the pressure differences due to the large movements

will be different from the indication of the derivatives in the Jacobin matrix.

Therefore, the movement of the last camber point on each strip should be

restrained around the initial position, when large movements are calculated

by the 3D search scheme. Another reason of restraining the movements of the

last camber points is to avoid abrupt changes of the extended trailing edge

and ensure that the trailing edge follows a smooth curve.

103

Figure 5.18: Pressure differences between the cut points on the initial and finalextensions.

r/R

100Gr

0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

Figure 5.19: Circulation on the final blade geometry.

104

The thrust/torque forces predicted by the present method, Pan (2009)

and experiment are plotted in Fig. 5.20 and Fig. 5.21. The results of the

present method show a better correlation with experimental measurements

than those in Pan (2009). In fact, the results of Pan (2009) had been improved

in Pan and Kinnas (2011) by using a full wake alignment. At this stage,

the combination of the improved method and the full wake alignment is not

available. Thus the same wake alignment in Pan (2009) is still used to study

the effects of the present method. The results by the present method are

expected to be improved when further coupled with the full wake alignment

used in Pan and Kinnas (2011).

Js

KT

0.7 0.75 0.8 0.85 0.9 0.95 1-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Experiment

Pan (2009)

Present method

Figure 5.20: KT of the propeller with finite thickness trailing edge.

105

Js

10KQ

0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

Experiment

Pan (2009)

Present method

Figure 5.21: KQ of the propeller with finite thickness trailing edge.

106

Chapter 6

Conclusions and Recommendations

6.1 Conclusions

In this thesis, a 3D viscous/inviscid interactive method has been de-

veloped, by coupling a 3D low-order potential based panel method and a 2D

integral boundary layer analysis method, for predicting the effects of fluid vis-

cosity on the performance of fully wetted propellers. The author follows the

assumption that the effects of the boundary layer along the span direction (ra-

dially outward of propeller blades) are small compared with those along the

stream wise direction (along constant radius of propeller blades). The assump-

tion allows one to simplify the 3D boundary layer into 2D on the strips along

the stream wise direction and to avoid solving the 3D integral boundary layer

equations. This method has been successfully applied to propeller, duct, and

hydrofoil cases and shows great correlations with experimental measurements

or RANS results. One significant improvement of this method, as compared

with previous work, is to further consider the viscous effects of neighboring

strips and blades.

Contributions of the present work include:

• The 3D viscous/inviscid interactive method considers the 3D boundary

107

layer on propeller blades or hydrofoils comprehensively, but with less

computation expense by applying a reasonable assumption. Previously,

a complete 3D viscous/inviscid interactive method by Milewski (1997)

was developed by coupling a panel method with the 3D integral bound-

ary layer equations. Such method requires large amounts of computer

resources and is difficult to be applied to complicated geometries such

propeller blades. Hufford (1992, 1994) and Sun (2008) assumed that the

3D boundary layer on propeller blades could be reduced into 2D on strips

along the constant radius. The author in this thesis follows their assump-

tion but further considers the viscous effects from neighboring strips and

blades by a 3D wall transpiration model. The effects of the neighboring

blades turns out to be very limited, but the effects of other strips lead

to obvious corrections of the pressure distributions on propeller blades.

• The present method can predict the friction on propeller blades more

accurately than empirical methods. Generally, such empirical methods

use empirical constants as the friction coefficients to take the effects of

fluid viscosity into account. The present method provides a better way

of predicting the skin friction coefficient. Through some cases in the

thesis, it is proven that the predicted friction coefficients by the present

method are much more reasonable than any empirical constant.

• The 3D viscous/inviscid interactive method is also applied to develop a

viscous image model. Singh (2009) developed an image model to simulate

108

the effects of slip walls. This model was developed based on a pertur-

bation potential based panel method and was limited only for inviscid

cases. A new viscous image model is developed in this thesis by cou-

pling the original model of Singh and the 3D viscous/inviscid interactive

method. The new model is applied to cases of wing blades between two

parallel slip walls, and the results show better correlation with RANS.

• The method proposed by Pan (2009,2011) for hydrofoils or propeller

blases with non-zero thickness trailing edges is improved. Panel methods

have some difficulties dealing with such trailing edges. This scheme

extends the trailing edge with finite thickness to form a sharp one for

panel methods. The geometry of the extension keeps changing until

two conditions, the no-lift and the pressure equivalence condition, on

the extension part are satisfied. The author in this thesis follows the

basic ideal of the method of Pan, but proposes a new extension and an

improved search scheme to make this method more reliable.

6.2 Recommendations

The present method produces accurate predictions of the effects of fluid

viscosity on propeller flows, but further improvements could be achieved in

future research.

• To improve the efficiency of calculations, the present 3D viscous/inviscid

interactive method only considers the boundary layer along the stream

109

wise direction, but ignores the one of crossing flow. Even though this is a

reasonable assumption, the boundary layer of the crossing flow may still

have some influences on the final results in some cases. In one instance,

for the DTMB Duct II case, the results predicted by the present method

are changed if the panels on the duct are given a large pitch angle. This

is because the panels with the pitch angle make the divided strips away

from the steam wise direction. In other words, the boundary layer is not

considered in the correct direction. In such situations, the effects of the

boundary layer on the other direction cannot be negligible.

• The method proposed by Pan (2009, 2011) for 2D hydrofoils with non-

zero thickness trailing edges has been improved and successfully applied

to two sample cases in 2D in the thesis. However, the study of 3D

cases is very limited. The search scheme for 3D blades should consider

the interactions among the strips by Newton’s method. The derivatives

used in the scheme can only be calculated numerically. Even though

the numerical derivatives of the sample case studied in this thesis seems

fine, the accuracy still cannot be guaranteed. Pan (2009, 2011) had to

sacrifice the efficiency of calculations and deal with the strips one by

one. Therefore, more studies and research should be conducted in the

future to make the method for 3D cases more reasonable and reliable. In

addition, the present method should be further coupled with a full wake

alignment to improve the predicted KT and KQ.

110

Appendix

111

Appendix 1

Evaluation of Influence Coefficients

In this appendix, the formulations of the influence coefficients for 2D

and 3D panels are presented.

1.1 2D Influence Coefficients

• Dipole

dij =1

2π

∫∂ ln r

∂ndS (1.1)

• Source

sij =1

2π

∫ln rdS (1.2)

1.2 3D Influence Coefficients

• Dipole

dij = − 1

4π

∫∂(1/r)

∂ndS (1.3)

• Source

sij = − 1

4π

∫1

rrdS (1.4)

112

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Vita

Xiangming Yu was born in Shenyang, Liaoning Province, China on

22 March 1987, the son of Xiaosheng Yu and Guixia Liu. He received the

degree of Bachelor of Science in Engineering from the department of Naval

Architecture and Ocean Engineering, Shanghai Jiao Tong University, in July,

2010. In the fall of 2010, he enrolled at the University of Texas at Austin for

graduate studies in the Ocean Engineering Group of the Department of Civil,

Architectural and Environmental Engineering.

Permanent address: Tiexi District, East Shenliao Road,Unit 47-3, apt 241Shenyang City, Liaoning Province 110000People’s Republic of China

This thesis was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a specialversion of Donald Knuth’s TEX Program.

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